Mathematics

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68
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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
  • 34,453
68
votes
5 answers

Number of onto functions

What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements. I found that if $m = 4$ and $n = 2$ the number of onto functions is $14$ . But is there a way to generalise this using a…
icyflame
  • 935
68
votes
12 answers

A way to find this shaded area without calculus?

This is a popular problem spreading around. Solve for the shaded reddish/orange area. (more precisely: the area in hex color #FF5600 ) $ABCD$ is a square with a side of $10$ , $APD$ and $CPD$ are semicircles, and $ADQB$ is a quarter circle. The…
Presh
  • 1,881
68
votes
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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68
votes
3 answers

Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

The equality $$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)\label{a}\tag{1}$$ follows from the fact that the sum of the first series is $\dfrac\pi4$ , whereas the sum of…
José Carlos Santos
  • 443,185
68
votes
10 answers

Is memory unimportant in doing mathematics?

The title says it all. I often heard people say something like memory is unimportant in doing mathematics. However, when I tried to solve mathematical problems, I often used known theorems whose proofs I forgot. EDIT Some of you may think that using…
Makoto Kato
  • 44,710
68
votes
3 answers

Approximating a $\sigma$-algebra by a generating algebra

Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$ . Then for all $B\in\cal B$ and $\varepsilon>0$ , we can find $A\in\cal A$ such that…
Davide Giraudo
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68
votes
6 answers

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm dx^n}f(x)=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k!…
Listing
  • 14,097
68
votes
6 answers

How to avoid perceived circularity when defining a formal language?

Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found here . What makes me uneasy about this definition is that words such as "set", "countable", "function", and…
IssaRice
  • 1,285
68
votes
15 answers

Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by $(-1)$ : $$\sqrt{x}=x-2$$ power of $2$ : $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $x=1$ is not a correct solution to the original equation. Why…
TheLogicGuy
  • 1,046
68
votes
12 answers

Easy way of memorizing values of sine, cosine, and tangent

My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday, but I may likely forget them after the test. So…
James Smith
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68
votes
5 answers

Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure

Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables formula of the form: $$ \int_{\phi(\Omega)} f d\lambda^m…
Oyqcb
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68
votes
16 answers

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2. In the second version of…
Rachel
  • 3,054
68
votes
5 answers

Sum of two closed sets in $\mathbb R$ is closed?

Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it? (By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in $Y$)…
ro44
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68
votes
1 answer

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square . $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me…
mathlove
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