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1500 questions
74
votes
4 answers
Are there any differences between tensors and multidimensional arrays?
I see lots of references saying things like
A tensor is a multidimensional or N-way array
But others that say things like
it should be remarked that other mathematical entities occur in physics that, like tensors, generally consist of…
rhombidodecahedron
- 1,582
74
votes
11 answers
Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)
Consider the following:
$1 = 1^2$
$2 + 2 = 2^2$
$3 + 3 + 3 = 3^2$
Therefore,
$\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$
Take the derivative of lhs and rhs and we get:
$\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}}…
user116
73
votes
3 answers
Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
I've found here the following integral.
$$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$
I've never seen it before and I also didn't find the evaluation on math.se. How could we…
user153012
- 12,890
- 5
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- 115
73
votes
9 answers
Subgroup of index $2$ is Normal
Please excuse the selfishness of the following question:
Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal.
Proof:
Because $|G:H|=2$, $G = H \cup aH$ for some $a \in G \setminus H$.
Let $x\in G$. Then $x \in H$ or $x…
William T.
- 787
73
votes
17 answers
What is a real world application of polynomial factoring?
The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this?
I feel a bit silly because it always bugged me when people asked that in…
Dan
- 871
73
votes
3 answers
Does non-symmetric positive definite matrix have positive eigenvalues?
I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.
Does this hold for non-symmetric matrices as well?
user19653
73
votes
3 answers
Mathematical research of Pokémon
In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some online servers let the players see the opponents…
Andrew Thompson
- 4,511
73
votes
3 answers
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this
$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$
$$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$
where $m$ is an integer.
$$\lim_{n\to\infty}n\sin(2\pi…
M. Amin
- 739
73
votes
6 answers
Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$
I am having trouble proving the statement:
Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon > 0$, the intersection of $S$ and $(0, \epsilon)$ is nonempty.
user11135
- 843
73
votes
3 answers
Distribution of the product of two (or more) uniform random variables
Say $X_1, X_2, \dots, X_n \sim U(0,1)$ are independent and identically distributed (i.i.d.) uniform random variables on the interval $(0,1)$. I am interested in finding the distribution of the product of $2$, $3$ or more such random variables,…
lulu
- 1,098
73
votes
2 answers
Can $18$ consecutive integers be separated into two groups such that their product is equal?
Can $18$ consecutive positive integers be separated into two groups such that their product is equal? We cannot leave out any number and neither we can take any number more than once.
My work:
When the smallest number is not $17$ or its…
Hawk
- 6,718
73
votes
5 answers
Matrices commute if and only if they share a common basis of eigenvectors?
I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
Yaroslav Bulatov
- 5,579
73
votes
0 answers
Dedekind Sum Congruences
For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} \mathfrak{s}( \tfrac{bc}{d}, \tfrac{a}{b^{\prime}…
user02138
- 17,314
73
votes
11 answers
$A_4$ has no subgroup of order $6$?
Can a kind algebraist offer an improvement to this sketch of a proof?
Show that $A_4$ has no subgroup of order 6.
Note, $|A_4|= 4!/2 =12$.
Suppose $A_4>H, |H|=6$.
Then $|A_4/H| = [A_4:H]=2$.
So $H \vartriangleleft A_4$ so consider the…
Stephen Cox
- 869
73
votes
6 answers
Similar matrices and field extensions
Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$?
Any help…
Melesia
- 731