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4
votes
4answers
430 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
1answer
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
0answers
325 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 ...
16
votes
2answers
976 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
4answers
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
1answer
330 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
1answer
316 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
2answers
552 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
2answers
582 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
1answer
296 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
8answers
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
1answer
227 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
4answers
802 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
0answers
189 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
0answers
236 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
1answer
425 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
0answers
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
3answers
921 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 ...
3
votes
0answers
239 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
9answers
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
4answers
764 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
3answers
700 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
1answer
515 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
2answers
508 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
0answers
532 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
1answer
214 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
7answers
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
2answers
850 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
0answers
632 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
2answers
994 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
0answers
94 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
2answers
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
0answers
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
3answers
768 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 ...
78
votes
19answers
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
5answers
856 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 ...
46
votes
5answers
4k 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
1answer
683 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
0answers
168 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
9answers
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 ...
14
votes
14answers
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 ...
18
votes
4answers
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 ...
23
votes
3answers
2k views

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 ...
8
votes
2answers
1k views

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 ...
23
votes
4answers
2k views

Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened. The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...
6
votes
10answers
842 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 ...
37
votes
14answers
3k views

A set for which it is hard to determine whether or not it is countable.

I got thinking recently, while trying to come up with a problem, that I did not know of any sets which were reasonable to define but for which it was very difficult to determine whether or not they ...
8
votes
3answers
839 views

P vs. NP resistant problems

According to Stephen Cook on wikipedia, http://en.wikipedia.org/wiki/P_versus_NP_problem ...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a ...
93
votes
14answers
12k 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
0answers
595 views

How many projects do you work on concurrently? [closed]

I was wondering how many concurrent research projects a typical math researcher works on at a given time. I ask because I currently have the oppertunity to start a second project on something I'm ...