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5 answers

If $e^A$ and $e^B$ commute, do $A$ and $B$ commute for finite dimensional matrices?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Additionally, what happens in $M_n(\mathbb{R})$? Nota Bene: As a…
Seirios
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66
votes
4 answers

Given a Fibonacci number , find the next Fibonacci number

The Fibonacci sequence is $0, 1, 1, 2, 3, 5, 8, 13, 21, 34,\ldots$, where each term after the first two is the sum of the two previous terms. Can we find the next Fibonacci number if we are given any Fibonacci number? For example, if $n = 8$ then…
ppSpp
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66
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7 answers

Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+x^3+x^2+x+1$$ what is the remainder of $P(x)$…
joeblack
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66
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7 answers

Could I be using proof by contradiction too much?

Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it directly, thereby reaching a "contradiction." Is this…
user64844
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66
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11 answers

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
nonuser
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66
votes
6 answers

What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample?…
liufu
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66
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9 answers

Prove every odd integer is the difference of two squares

I know that I should use the definition of an odd integer ($2k+1$), but that's about it. Thanks in advance!
papercuts
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66
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1 answer

when can we interchange integration and differentiation

Let $f$ be a Riemann Integrable function over $\mathbb{R}^2$. When can we do this? $$\frac{\partial}{\partial\theta}\int_{a}^{b}f(x,\theta)dx=\int_{a}^{b}\frac{\partial}{\partial\theta}f(x,\theta)dx$$ (Here, $a$ and $b$ are not a function of…
user467365
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66
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2 answers

Conway's "Murder Weapon"

The following quote is an excerpt from an interview with John Conway: Coxeter came to Cambridge and he gave a lecture, then he had this problem for which he gave proofs for selected examples, and he asked for a unified proof. I left the…
Oiler
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66
votes
26 answers

What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices. Preferably these examples would be…
providence
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7 answers

$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee

So one of the question on the MIT Integration Bee has baffled me all day today $$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$$ I have tried a variety of things to do this, starting with Integration By Parts Part 1 $$\frac{\tan…
Teh Rod
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66
votes
2 answers

What is the integral of 1/x?

What is the integral of $\frac{1}{x}$? Do you get $\ln(x)$ or $\ln|x|$? In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$? Also, what is the derivative of $|f(x)|$? Is it $f'(x)$ or $|f'(x)|$?
hollow7
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66
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9 answers

What's wrong with this reasoning that $\frac{\infty}{\infty}=0$?

$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$ You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus…
JobHunter69
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66
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18 answers

Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every 3rd term as positive, thus it would…
Kristaps John Balodis
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66
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2 answers

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$ where $n\#$ is a primorial, and $p_k$…
Timmy Turner
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