# Tagged Questions

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### 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 - ...
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### “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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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( ...
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### 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 ...
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### 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 ...
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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,...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
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### Where does Aphex Twin's “windowlicker” equation come from? [closed]

$\Delta M_i^{-1} = -\alpha \sum\limits_{n=1}^N D_i [n] \left[\sum\limits_{j \in C[i]} F_{ji} [n-1] + Fext_i [n^{-1}]\right]$ This is the name of the second song on Aphex Twin's album "Windowlicker". ...
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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 ...
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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 ...
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### Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
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### Are there uncountably many essentially inequivalent versions of Mathematics?

Hi everyone, Disclaimer 1: logic and set theory are a long way from my field, so apologies in advance if I demonstrate extreme ignorance or stupidity, and please correct me if (when?) I write stupid ...
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### Why semigroups are important? [closed]

There is known a lot about semigroups, mostly about their inner structure etc. But is there any use of semigroups in the general Maths -- like that of groups?
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### Is there a conjunction bias?

This is slightly related to question The unprecedented success of the “intersection” operator . Apart from a set of maths books of null measure, most have the following property: Objects ...
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### Advice on Giving a Talk [closed]

What advice do you have for giving a talk on a mathematical research paper to people in other fields in science (not physics nor astronomy) but without lot of math background? Thanks.
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### Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? [closed]

Author : Z.A. Melzak Book Title : Companion to Concrete Mathematics. Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976. I found this book extremely nice. To whet your appetite ...
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Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: $\... 1answer 1k views ### Unpopular “elementary” theorems/identities to impress an audience of mathematicians. [closed] This question grew out of my recent job interview. Since the interviewers were math professors, I had a hard time searching for interesting elementary theorems in case I got asked for one. I thought ... 6answers 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 ... 94answers 12k views ### What would you want to see at the Museum of Mathematics? EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ... 1answer 1k views ### Technical trends quietly aimed at big open problems? [closed] When I was an undergraduate 35 years ago, I made the mistake of asking some of my mathematics professors what well-known open problems they liked to think about. I got the message that this was ... 11answers 3k views ### 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 ... 16answers 2k views ### 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; ... 0answers 166 views ### Regarding a Feature of Multivariate Real Function Any real function can be expressed as a function of the sum of two monotonic real functions? More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ... 0answers 584 views ### What is the name for$(a^2 + b^2 + c^2 +…)/(a + b + c +…)\$? [closed]

That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.
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### Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
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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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### 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 ...