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1500 questions
130
votes
7 answers
Are all algebraic integers with absolute value 1 roots of unity?
If we have an algebraic number $\alpha$ with (complex) absolute value $1$, it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$). For example, $(3/5 + 4/5 i)$ is not a root of unity.
But if we assume that…
Jonas Kibelbek
- 7,380
130
votes
0 answers
On the Constant Rank Theorem and the Frobenius Theorem for differential equations.
Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the Implicit Function Theorem (finite-dimensional vector…
Elias Costa
- 15,282
130
votes
16 answers
Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$
$x,y,z >0$, prove
$$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$
Note:
Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem…
HN_NH
- 4,511
130
votes
1 answer
Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?
Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to construct such a quartic and rule out the existence of…
Milo Brandt
- 61,938
129
votes
1 answer
Motivation of irrationality measure
I have a question about the irrationality of $e$:
In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$$ So the series for $e^{-1}$ is…
Damien
- 1
129
votes
19 answers
Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"
Are there any…
Mike Spivey
- 56,818
129
votes
7 answers
Is there a function that grows faster than exponentially but slower than a factorial?
In big-O notation the complexity class $O(2^n)$ is named "exponential". The complexity class $O(n!)$ is named "factorial".
I believe that $f(n) = O(2^n)$ and $g(n) = O(n!)$ means that $\dfrac{f(n)}{g(n)}$ goes to zero in the limit as $n$ goes to…
Robert L. Read
- 1,217
129
votes
3 answers
All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$
I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$.
I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with infinite periods denoted by brackets.
$$2=\sqrt{2 +…
Yuriy S
- 32,728
128
votes
14 answers
Good abstract algebra books for self study
Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study…
sxd
- 3,534
128
votes
26 answers
Different ways to prove there are infinitely many primes?
This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more…
Patrick Da Silva
- 42,548
128
votes
8 answers
Are the "proofs by contradiction" weaker than other proofs?
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by…
AgCl
- 6,532
128
votes
6 answers
Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)?
Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").
Yet no one uses these. Why is this?
thecat
- 1,878
128
votes
2 answers
Why is it hard to prove whether $\pi+e$ is an irrational number?
From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?"
Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…
users31526
- 2,077
128
votes
1 answer
Simplicial Complex vs Delta Complex vs CW Complex
I am a little confused about what exactly are the difference(s) between simplicial complex, $\Delta$-complex, and CW Complex.
What I roughly understand is that $\Delta$-complexes are generalisation of simplicial complexes (without the requirement…
yoyostein
- 20,428
127
votes
8 answers
derivative of cost function for Logistic Regression
I am going over the lectures on Machine Learning at Coursera.
I am struggling with the following. How can the partial derivative of
$$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+(1-y^{i})\log(1-h_\theta(x^{i}))$$
where…
dreamwalker
- 1,415