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1500 questions
223
votes
6 answers
Why does this matrix give the derivative of a function?
I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$
And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial,…
ASKASK
- 9,000
- 4
- 28
- 49
222
votes
13 answers
Inverse of the sum of matrices
I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-1}$ is not known, but if it is necessary then it…
Tomek Tarczynski
- 2,974
221
votes
7 answers
$L^p$ and $L^q$ space inclusion
Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied?
Confused
- 2,219
221
votes
16 answers
Optimizing response times of an ambulance corp: short-term versus average
Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. There is a belief that this results in the fastest…
Laurence
- 1,483
221
votes
18 answers
Do men or women have more brothers?
Do men or women have more brothers?
I think women have more as no man can be his own brother. But how one can prove it rigorously?
I am going to suggest some reasonable background assumptions:
There are a large number of individuals, of whom half…
layman
- 1,877
218
votes
5 answers
What do modern-day analysts actually do?
In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about the classification of finite simple groups, and…
Jesse Madnick
- 31,524
218
votes
4 answers
Why can a Venn diagram for $4+$ sets not be constructed using circles?
This page gives a few examples of Venn diagrams for $4$ sets. Some examples:
Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could…
Larry Wang
- 9,513
217
votes
21 answers
Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
Jichao
- 8,008
217
votes
13 answers
Why is compactness so important?
I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from…
FireGarden
- 5,835
216
votes
5 answers
What is the geometric interpretation of the transpose?
I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example,…
Zed
- 2,161
215
votes
13 answers
Is computer science a branch of mathematics?
I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about computer science/programming that are very…
user107952
- 20,508
213
votes
22 answers
Is it faster to count to the infinite going one by one or two by two?
A child asked me this question yesterday:
Would it be faster to count to the infinite going one by one or two by two?
And I was split with two answers:
In both case it will take an infinite time.
Skipping half of the number should be really…
Thomas Ayoub
- 1,635
- 3
- 12
- 12
213
votes
5 answers
The sum of an uncountable number of positive numbers
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).
Proof: Let…
Benji
- 5,880
212
votes
15 answers
Identification of a quadrilateral as a trapezoid, rectangle, or square
Yesterday I was tutoring a student, and the following question arose (number 76):
My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel,…
The Chaz 2.0
- 10,464
211
votes
7 answers
Transpose of inverse vs inverse of transpose
Given a square matrix,
is the transpose of the inverse equal to the inverse of the transpose?
$$
(A^{-1})^T = (A^T)^{-1}
$$
Void Star
- 2,555