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1500 questions
86
votes
4 answers
Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
Is the following true?
Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a
further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
I guess that it is true but I am not sure how to prove…
gulim
- 903
86
votes
3 answers
Meaning of “arg min”
Would someone be so kind to explain this to me:
$$\pi_nk=\left\{\begin{array}{cl}1&\textrm{if }k=\arg\min_j\left\Vert\mathbf x_n-\mu_j\right\Vert^2\\0&\textrm{otherwise}\end{array}\right..$$
Especially the $\arg\min$ part.
(It's from the $k$-means…
Olivier_s_j
- 1,025
86
votes
9 answers
In Linear Algebra, what is a vector?
I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is.
I know in physics a vector is a geometric object that has a magnitude and a…
Paul Lee
- 1,015
86
votes
9 answers
What is the meaning of the third derivative of a function at a point
(Originally asked on MO by AJAY.)
What is the geometric, physical, or other meaning of the third derivative of a function at a point?
If you have interesting things to say about the meaning of the first and second derivatives, please do so.
Gil Kalai
- 1,143
86
votes
5 answers
Why is a circle in a plane surrounded by 6 other circles?
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers?
I'm looking for an answer deeper than "there are…
John Smith
- 2,280
86
votes
4 answers
A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$
While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because the integral looks very interesting to me.…
Venus
- 10,966
85
votes
8 answers
What is integration by parts, really?
Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher derivatives of a function into information about an…
Elle Najt
- 20,740
85
votes
8 answers
Difference between continuity and uniform continuity
I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book defines continuity as:
Definition 4.3.1. A…
user124005
- 863
85
votes
3 answers
What algorithm is used by computers to calculate logarithms?
I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware.
So the question is: what…
zar
- 4,602
85
votes
3 answers
Fastest way to check if $x^y > y^x$?
What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that?
The issue is that $x$ and $y$ can be very large.
learner
- 905
85
votes
4 answers
difference between dot product and inner product
I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the integral of the product of $2$ functions is…
CuriousGeorge119
- 1,207
85
votes
6 answers
What is the intuition behind uniform continuity?
There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it.
Playing with the definition,…
Spencer Kraisler
- 2,882
85
votes
7 answers
Quotient ring of Gaussian integers
A very basic ring theory question, which I am not able to solve. How does one show that
$\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$.
Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are…
user9413
85
votes
7 answers
Understanding Borel sets
I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just…
Tikhon Belousko
- 1,221
85
votes
8 answers
What is "Bra" and "Ket" notation and how does it relate to Hilbert spaces?
This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my textbook (Griffiths) and read countless of pdf's on the…
qmd
- 4,275