Mathematics

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1500 questions
147
votes
5 answers

Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$

How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$ ? This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I am unsure how they are relevant.
jaynp
  • 2,251
147
votes
6 answers

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career…
Potato
  • 41,689
147
votes
0 answers

Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian Art Problem . For…
Ed Pegg
  • 22,188
146
votes
2 answers

Is there any mathematical reason for this "digit-repetition-show"?

The number $$\sqrt{308642}$$ has a crazy decimal representation : $$555.5555777777773333333511111102222222719999970133335210666544640008\cdots $$ Is there any mathematical reason for so many repetitions of the digits ? A long block containing only a…
Peter
  • 87,158
145
votes
6 answers

Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A…
user1736
  • 9,083
145
votes
0 answers

Probability for an $n\times n$ matrix to have only real eigenvalues

Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$ . What is the probability that $A$ has only real eigenvalues? The answer cannot be $0$ or $1$ , since the set of matrices with distinct real…
Exodd
  • 12,331
145
votes
10 answers

Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red. Which area is greater? Let the area of each large…
Mr Pie
  • 9,794
145
votes
16 answers

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums . Can someone provide a nice proof that $$A(1,1) =…
Mike Spivey
  • 57,148
145
votes
6 answers

Why is integration so much harder than differentiation?

If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in…
Venge
  • 1,631
145
votes
9 answers

What exactly is the difference between a derivative and a total derivative?

I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w / \partial r$ and $\partial w / \partial…
Chibueze Opata
  • 1,585
144
votes
18 answers

How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?

So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it: Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen. I tried to solve it and failed! Does anybody know how to solve…
zerosofthezeta
  • 5,720
144
votes
1 answer

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where $p_0 > 0$ and $p_{k+1} > p_k$ for all $k$ . In…
Yuval Filmus
  • 58,203
144
votes
23 answers

Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a…
Anonymous
  • 281
144
votes
8 answers

Prove that $||x|-|y||\le |x-y|$

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove…
Anonymous
  • 2,566
143
votes
30 answers

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone…
Michiel Van Couwenberghe
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