Most Popular
1500 questions
145
votes
11 answers
Physical meaning of the null space of a matrix
What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank $r$ of a matrix as the minimum number…
user541686
- 13,772
145
votes
14 answers
Why do units (from physics) behave like numbers?
What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view?
I've made the observation that units "behave like numbers". For example, we can divide them (as in $m/s$, which is a unit of speed), and also square…
user377104
145
votes
6 answers
Why does mathematical convention deal so ineptly with multisets?
Many statements of mathematics are phrased most naturally in terms of multisets. For example:
Every positive integer can be uniquely expressed as the product of a multiset of primes.
But this theorem is usually phrased more clumsily, without…
MJD
- 65,394
- 39
- 298
- 580
144
votes
2 answers
Is there a known well ordering of the reals?
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on…
Seamus
- 4,005
144
votes
18 answers
How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?
So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it:
Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen.
I tried to solve it and failed! Does anybody know how to solve…
zerosofthezeta
- 5,646
144
votes
4 answers
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that if a function is in two $L^p$ spaces, (e.g. $p_1$…
user1736
- 8,573
144
votes
5 answers
Can someone clearly explain about the lim sup and lim inf?
Can some explain the lim sup and lim inf?
In my text book the definition of these two is this.
Let $(s_n)$ be a sequence in $\mathbb{R}$. We define
$$\lim \sup\ s_n = \lim_{N \rightarrow \infty} \sup\{s_n:n>N\}$$
and
$$\lim\inf\ s_n =…
eChung00
- 2,963
- 8
- 29
- 42
144
votes
1 answer
Identification of a curious function
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},
$$
where $p_0 > 0$ and $p_{k+1} > p_k$ for all $k$. In…
Yuval Filmus
- 57,157
144
votes
3 answers
How to find the Galois group of a polynomial?
I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the technical stuff. A specific example would be how…
IBS
- 4,155
144
votes
23 answers
The Best of Dover Books (a.k.a the best cheap mathematical texts)
Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books are not: For example, while something like…
Three
- 852
- 3
- 10
- 13
144
votes
10 answers
Is the blue area greater than the red area?
Problem:
A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red.
Which area is greater?
Let the area of each…
Mr Pie
- 9,459
144
votes
10 answers
When is matrix multiplication commutative?
I know that matrix multiplication in general is not commutative. So, in general:
$A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$
But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix $\forall B \in…
Martin Thoma
- 9,821
144
votes
6 answers
lim sup and lim inf of sequence of sets.
I was wondering if someone would be so kind to provide a very simple explanation of $\limsup$ and $\liminf$ of a sequence of sets. For a sequence of subsets $A_n$ of a set $X$ we have $$\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n…
Comic Book Guy
- 5,748
143
votes
22 answers
List of interesting math podcasts?
mathfactor is one I listen to. Does anyone else have a recommendation?
Tim
- 119
143
votes
5 answers
On Ph.D. Qualifying Exams
Where can I find Ph.D. qualifying exams questions? Is there any website that keeps a collection of such problems?
I need it for doing some revision of the basic topics. I know of a book but that doesn't have the full collection.
Koushik
- 4,472