**49**

votes

**3**answers

5k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**41**

votes

**6**answers

11k views

### Best tablet computer for mathematics [closed]

I'm not sure if this is completely appropriate, but I thought I'd ask here.
I'm in the market for a tablet computer. Unfortunately, my (mathematical) needs are very different from the needs of the ...

**35**

votes

**37**answers

30k views

### What is your favorite “strange” function? [closed]

There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you ...

**35**

votes

**6**answers

3k views

### What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...

**35**

votes

**8**answers

3k views

### The shortest path in first passage percolation

Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and differ by one in the other.)
...

**35**

votes

**2**answers

1k views

### Moving one family of commuting self-adjoint operators to another without losing commutativity on the way

This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...

**34**

votes

**1**answer

1k views

### Mr. G.P.K.'s questions [closed]

WARNING: An acquaintance of mine, Mr.Goosepond Prhklstr Kratchinabritchisitch, has requested permission to post his questions under my username. When I asked him why he didn't do it under his own ...

**32**

votes

**0**answers

1k views

### Two-convexity ⇒ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

**31**

votes

**5**answers

2k views

### Can pure mathematics harness citizen science?

Having just finished Michael Nielsen's book "Reinventing Discovery", I find myself wondering if there are ways that pure mathematics research can engage the public in the way that GalaxyZoo or Foldit ...

**31**

votes

**2**answers

2k views

### Gently falling functions

I wonder if it is possible to characterize the class of
gently falling functions, which I would like to define
as follows.
Let $g(x)$ be a $C^2$ function defined on an interval
$R \subseteq \mathbb{R}$...

**29**

votes

**9**answers

3k views

### Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...

**25**

votes

**14**answers

3k views

### Making sure that you have comprehended a concept

Hi,
I have a question that I've been thinking about for a long time.
How can you assure yourself that you've fully comprehended a concept or the true meaning of a theorem in mathematics?
I mean how ...

**25**

votes

**2**answers

1k views

### Is there a finite family of functions such that the max of any two functions can be dominated by a third?

Is it true that for every $t$ there is an $n$ and there exists a finite function
family, $\cal F$, whose members are from $[n] \to \mathbb N$ (taking all different
values) and for any $f_1, \ldots, ...

**24**

votes

**2**answers

1k views

### Euler characteristic and universal cover

Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...

**23**

votes

**1**answer

2k views

### Why do we use $\epsilon$ and $\delta$?

My understanding (from a talk by Rob Bradley) is that Cauchy is responsible for
the now-standard $\epsilon{-}\delta$ formulation of calculus, introduced in his
1821 Cours d’analyse. Although perhaps ...

**22**

votes

**16**answers

3k views

### functions satisfying “one-one iff onto”

Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...

**22**

votes

**4**answers

2k views

### Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.
How can we directly visualize how and in what sense the Cech cohomology of a cover does this?
In ...

**21**

votes

**2**answers

1k views

### Proving non-existence of solutions to $3^n-2^m=t$ without using congruences

I made a passing comment under Max Alekseyev's cute answer to this question and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was ...

**21**

votes

**1**answer

1k views

### Red-blue alternating paths

Suppose we have two simple graphs on the same vertex set. We will call one of them red, the other blue. Suppose that for $i=1,..,k$ we have $deg (v_i)\ge i$ in both graphs, where $V_k=\{v_1,\ldots,v_k\...

**20**

votes

**11**answers

5k views

### The role of the mean value theorem (MVT) in first-year calculus.

Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...

**20**

votes

**2**answers

2k views

### What is the L-function version of quadratic reciprocity?

Quadratic reciprocity theorems states that for two different odd prime p and q,
we have (p/q)(q/p)=(-1)^(p-1)(q-1)/4.
What is the statement of this theorem in L-function?

**20**

votes

**6**answers

2k views

### Formal consequences of Riemann-Roch (multiple answers welcome)

This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...

**19**

votes

**6**answers

2k views

### Generalizations of “standard” calculus

We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...

**18**

votes

**9**answers

5k views

### How to motivate and present epsilon-delta proofs to undergraduates?

This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...

**18**

votes

**13**answers

3k views

### Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? [closed]

So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute $...

**18**

votes

**4**answers

2k views

### The “ds” which appears in an integral with respect to arclength is not a 1-form. What is it?

The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear. So $ds$ is not a 1-form. It ...

**18**

votes

**0**answers

601 views

### Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is:
Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that
$$ \binom{2n}{n} \equiv 2\pmod p ? $$
...

**17**

votes

**8**answers

10k views

### Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...

**16**

votes

**6**answers

7k views

### Angle Maximizing the Distance of a Projectile

It is well-known that to maximize the horizontal distance traveled by a projectile fired from the ground at a given speed, one should fire it at a $45^\circ$ angle. What's less-known, though not too ...

**16**

votes

**5**answers

1k views

### Homological algebra and calculus (as in Newton)

This question reminded me of a possibly stupid idea that I had a while back.
On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely ...

**16**

votes

**2**answers

1k views

### There is mathematics behind the 1989 Tour de France !

The $1989$ Tour was won by Greg Lemond (USA, $1961$ - ), who beat Laurent Fignon (France, $1960$ - $2010$) by $8''$. Yes, eight seconds! The closest tour in history.
Let me recall a few rules ...

**16**

votes

**1**answer

504 views

### Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.
Is $S\times S$ homeomorphic to $S$?
By Luzin ...

**15**

votes

**5**answers

3k views

### Where to publish a paper on the Mafia game?

I wrote a research paper "A mathematical model of the Mafia game" (arXiv:1009.1031 [math.PR]). However, I do not know where to publish it. As an undergraduate studying majorly physics, I have little ...

**15**

votes

**6**answers

5k views

### What's the notation for a function restricted to a subset of the codomain?

Suppose I have a function f : A → B between two sets A and B. (The same question applies to group homomorphisms, continuous maps between topological spaces, etc. But for simpicity let's restrict ...

**15**

votes

**3**answers

2k views

### Model category structure on Set without axiom of choice

There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...

**15**

votes

**0**answers

793 views

### Optimal Monotone Families for the Discrete Isoperimetric Inequality

Background: the Discrete Isoperimetric Inequality
Start with a set X={1,2,...,n} of n elements and the family $2^X$ of all subsets of X.
For a real number p between zero and one, we consider a ...

**14**

votes

**2**answers

2k views

### The non-traveling mathematician problem

This is a career question. I have just begun a research postdoc position in Southern California. It has been hard, but I've enjoyed teaching my first graduate courses and working on research and ...

**14**

votes

**5**answers

1k views

### Characterizing visual proofs

``Proofs without words'' is a popular column in the Mathematics magazine.
Question: What would be a nice way to characterize which assertions have such visual proofs? What definitions would one need?...

**14**

votes

**1**answer

1k views

### Probability of a Point on a Unit Sphere lying within a Cube

Suppose we have a (n-1 dimensional) Unit Sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$
What is the probability that a randomly selected point on the sphere, $ (x_1,x_2,x_3,...,x_n)$, ...

**14**

votes

**2**answers

593 views

### Curve integral of exponent of superharmonic function.

Let $\phi$ be a real smooth superharmonic function on unit disc $D$ in $\mathbb C$; i.e. $\triangle \phi\le 0$.
Then there is a curve $\gamma$ from the center of $D$ to its boundary such that
$$\...

**13**

votes

**13**answers

2k views

### Applications of connectedness

In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea ...

**13**

votes

**3**answers

583 views

### Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following:
Given some simplicial complex $M$ with rational vertices representing a closed manifold,
can one decide (in finite time) if $M$ is ...

**13**

votes

**5**answers

7k views

### If d/dx is an operator, on what does it operate?

If $\frac{d}{dx}$ is a differential operator, what are its inputs? If the answer is "(differentiable) functions" (i.e., variable-agnostic sets of ordered pairs), we have difficulty distinguishing ...

**13**

votes

**0**answers

515 views

### Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...

**12**

votes

**5**answers

1k views

### What is $\sum (x+\mathbb{Z})^{-2}$?

This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...

**12**

votes

**4**answers

1k views

### Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...

**12**

votes

**2**answers

631 views

### Independence of Leibniz rule and locality from other properties of the derivative?

The following is meant to be an axiomatization of differential calculus of a single variable. To avoid complications, let's say that $f$, $g$, $f'$, and $g'$ are smooth functions from $\mathbb{R}$ to $...

**12**

votes

**2**answers

913 views

### Hopf Algebra for a physicist

Hello,
for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...

**12**

votes

**1**answer

2k views

### Anything going on for a mathematician stuck at New York?

First of all, apologies for the really non-standard question/announcement. I know this is not what MO was intended for, but in this situation it is the easiest way to reach (perhaps) the right person.
...

**12**

votes

**1**answer

367 views

### Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...