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1500 questions
68
votes
4 answers
why geometric multiplicity is bounded by algebraic multiplicity?
Define
The algebraic multiplicity of $\lambda_{i}$ to be the degree of the root $\lambda_i$ in the polynomial $\det(A-\lambda I)$.
The geometric multiplicity the be the dimension of the eigenspace associated with the eigenvalue $\lambda_i$.
For…
Jack2019
- 1,605
68
votes
2 answers
Is this similarity to the Fourier transform of the von Mangoldt function real?
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$
The von Mangoldt function should then be:
$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n}…
Mats Granvik
- 7,614
68
votes
1 answer
Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.
Introduction:
We know that:
$$\sum_{x=0}^\infty \frac{1}{x!}=e$$
But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number? This interestingly is just a multiple of $e$ and an Incomplete…
Тyma Gaidash
- 13,576
68
votes
28 answers
Is there a great mathematical example for a 12-year-old?
I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets.
Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and…
Mark Bennet
- 101,769
68
votes
2 answers
Why doesn't Cantor's diagonal argument also apply to natural numbers?
In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string.
My question is: why can't we begin by representing each natural number as an infinite bit string? So that 0 =…
usul
- 4,084
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
- 1,861
- 2
- 18
- 25
68
votes
4 answers
What is a differential form?
can someone please informally (but intuitively) explain what "differential form" mean? I know that there is (of course) some formalism behind it - definition and possible operations with differential forms, but what is the motivation of introducing…
Check drummer
- 803
68
votes
10 answers
A linear operator commuting with all such operators is a scalar multiple of the identity.
The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.
We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in…
abeln
- 615
68
votes
4 answers
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1…
user55114
68
votes
5 answers
Help understanding Algebraic Geometry
I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with algebraic or projective geometry.
I'm wondering if…
user25470
- 1,073
- 2
- 10
- 15
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…
Makoto Kato
- 44,216
68
votes
3 answers
In combinatorics, how can one verify that one has counted correctly?
This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly.
It's easiest to explain with an example, so I'll give…
Stephen
- 1,063
68
votes
4 answers
Intersection of finite number of compact sets is compact?
Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true.
I said that this is true because the intersection of finite number of compact sets are closed. Which…
John Buchta
- 691
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
- 181,608