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1500 questions
112
votes
2 answers
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
Let $1\leq p < \infty$ . Suppose that $\{f_k, f\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$ . Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$ A…
user1736
- 9,083
112
votes
22 answers
Why is there no "remainder" in multiplication
With division, you can have a remainder (such as $5/2=2$ remainder $1$). Now my six year old son has asked me "Why is there no remainder with multiplication"? The obvious answer is "because it wouldn't make sense" or just "because". Somewhat I have…
topskip
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112
votes
3 answers
Grothendieck 's question - any update?
I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first conversations with me, he raised the question (asked of…
Bombyx mori
- 20,238
112
votes
13 answers
Why is "the set of all sets" a paradox, in layman's terms?
I've heard of some other paradoxes involving sets (i.e., "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "the set of all sets" a paradox? It seems like it…
Justin L.
- 15,298
112
votes
8 answers
Prove the theorem on analytic geometry in the picture.
I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the discussion. Edit: Current progress: The theorem…
user122049
- 1,652
111
votes
4 answers
Does this property characterize a space as Hausdorff?
As a result of this question , I've been thinking about the following condition on a topological space $Y$: For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$…
Arturo Magidin
- 420,020
111
votes
12 answers
Logic puzzle: Which octopus is telling the truth?
King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth. One day, four servants met. The blue one says, “Altogether, we have 28 legs.”…
a1bcdef
- 2,335
111
votes
3 answers
How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
Nice problem: Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that $$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$ where $\lfloor x \rfloor$ is the…
math110
- 95,328
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- 156
- 531
111
votes
6 answers
Continuous bijection from $(0,1)$ to $[0,1]$
Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
Alex
- 1,377
111
votes
6 answers
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind of equation? If this equation doesn't have any…
hasnohat
- 3,972
111
votes
2 answers
A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$
The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of composition, not known to have an identity or inverses.…
Lubin
- 65,809
111
votes
5 answers
Factorial and exponential dual identities
There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I vaguely remember a generating function/integration…
Mitch
- 8,847
111
votes
1 answer
Tate conjecture for Fermat varieties
I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here ) and have a question about his work on (conjecturing!) the Tate conjecture for the Fermat variety $X_m^r$, defined by the…
Alex
- 4,188
111
votes
8 answers
What makes elementary functions elementary?
Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, logarithmic, exponential, and $n$th roots, and solving…
user23784
110
votes
24 answers
Open mathematical questions for which we really, really have no idea what the answer is
There is no shortage of open problems in mathematics . While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both directions but rather have a pretty clear idea of…
user139000