## Top new questions this week:

### Is there something special about 2015?

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...

(algebra-precalculus) (elementary-number-theory) (recreational-mathematics)
 asked by Joonas Ilmavirta 37 votes
 answered by Jack D'Aurizio 64 votes

### An easy example of a non-constructive proof without an obvious "fix"?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...

(logic) (examples-counterexamples) (constructive-mathematics)
 asked by Valentin Golev 33 votes
 answered by Neil Strickland 56 votes

### Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...

(differential-equations) (recurrence-relations) (generating-functions) (laplace-transform)
 asked by oldrinb 19 votes
 answered by maxerize 3 votes

### Numbers that are divisible by the number of primes smaller than them

Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ...

(number-theory) (prime-numbers)
 asked by barak manos 14 votes
 answered by Nacho Darago 11 votes

### Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...

(probability) (statistics) (applications) (law-of-large-numbers)
 asked by user1891836 13 votes
 answered by Erick Wong 17 votes

### How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...

(calculus) (integration) (proof-writing) (definition)
 asked by goblin 13 votes
 answered by Jack M 9 votes

### Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty)$ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$

(real-analysis) (limits) (continuity) (examples-counterexamples)
 asked by tone 13 votes
 answered by Hagen von Eitzen 24 votes

## Greatest hits from previous weeks:

### Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality ...

(elementary-number-theory) (divisibility) (puzzle) (problem-solving)
 asked by RAO 62 votes
 answered by Aaron Hall 36 votes

### Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...

(game-theory) (fair-division)
 asked by VividD 413 votes
 answered by Chris Culter 173 votes

## Can you answer these?

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...

(probability) (statistics) (algebraic-geometry) (differential-geometry) (random-matrices)
 asked by Peter 4 votes

### Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...

(geometry) (algebraic-geometry)
 asked by Chao Xu 7 votes

### Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...

(differential-equations)
 asked by Joans 4 votes

## New blog post:

### Welcome the new trio of moderators of 2014

by Community Blog on Jan 26

The MSE elections are over. Sure, this is a bit late, but that’s no excuse not to be excited. Welcome the new moderators! We should also take a moment to thank the retiring moderators Alex ...

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