Christian Blatter
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Connection between Fourier transform and Taylor series
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146 votes

Assume that the Taylor expansion $f(x)=\sum_{k=0}^\infty a_k x^k$ is convergent for some $|x|>1$. Then $f$ can be extended in a natural way into the complex domain by writing $f(z)=\sum_{k=0}^\...

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Intuitive interpretation of the Laplacian Operator
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129 votes

Assume the function $f$ is $C^2$ in a neighbourhood of ${\bf 0}\in{\mathbb R}^n$. Using the Taylor expansion of $f$ at ${\bf 0}$ one can prove the following: $$\Delta f({\bf 0}) =\lim_{r\to 0+}\ {2n\...

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20 circles in the plane, all passing through the origin
96 votes

A hint: Move the origin to $\infty$ using the map $z\mapsto{1\over z}$. Then the circles become lines, no two of them parallel, and no three of them going through the same point. Denote the number ...

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Is there a clever solution to Arnold's "merchant problem"?
85 votes

At the end the tea cup is as full as at the start. This implies that the added wine is exactly outweighed by the tea that has disappeared.

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What should be the intuition when working with compactness?
70 votes

You may read various descriptions and consequences of compactness here. But be aware that compactness is a very subtle finiteness concept. The definitive codification of this concept is a fundamental ...

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What does the term "undefined" actually mean?
64 votes

To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b ...

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Is math built on assumptions?
60 votes

The basic assumption of the "working mathematician" is the following: The logical and set-theoretical environment of his deliberations is not contradictory. When we say: "Assume the triangle $ABC$ ...

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New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$
49 votes

$${1\over(1-x)^2}={1\over 1-x}\cdot{1\over 1-x}=\sum_{j\geq0} x^j\cdot\sum_{k\geq0}x^k =\sum_{r\geq0} x^r\left(\sum_{j+k=r}1\right)=\sum_{r\geq0}(r+1)x^r\ .$$

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Is there a vector space that cannot be an inner product space?
48 votes

I'm assuming the ground field is ${\mathbb R}$ or ${\mathbb C}$, because otherwise it's not clear what an "inner product space" is. Now any vector space $X$ over ${\mathbb R}$ or ${\mathbb C}$ has a ...

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What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
47 votes

For given $n\geq2$ one has $$e\cdot n!=n!\sum_{k=0}^\infty{1\over k!}=n!\left(\sum_{k=0}^n{1\over k!}+\sum_{k=n+1}^\infty{1\over k!}\right)=m_n+r_n$$ with $m_n\in{\mathbb Z}$ and $${1\over n+1}<r_n=...

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Representing the multiplication of two numbers on the real line
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45 votes

You can do so, but if you want to represent the result of the multiplication again as a length you have to choose a unit. The ancient Greeks didn't come up with this idea; whence their products were ...

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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
45 votes

Consider the function $f(t):=t^2\ \ (-\pi\leq t\leq \pi)$, extended to all of ${\mathbb R}$ periodically with period $2\pi$. Developping $f$ into a Fourier series we get $$t^2 ={\pi^2\over3}+\sum_{k=...

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Determine whether $f(x)={\sin x \over x}$ is uniformly continuous in $\mathbb R$
43 votes

One has $$f(x):={\sin x\over x}=\int_0^1 \cos(\tau x)\ d\tau\ .$$ It follows that $$|f(x)-f(y)|\leq \int_0^1|\cos(\tau x)-\cos(\tau y)|\ d\tau\leq \int_0^1\tau |x-y|\ d\tau={1\over2}|x-y|\ .$$ This ...

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Is 1100 a valid state for this machine?
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43 votes

A general advice: Whenever you have to distill a simple answer from a confusing medley of cases, look out for the invariant! The remainder mod 6 increases by $2$ per hour; therefore it is periodic ...

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Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
41 votes

Take $1$ and any square $>1$. Multiply both of them with the smallest common denominator of their AM and HM.

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What is the importance of the Collatz conjecture?
40 votes

So many mathematicians, and famous ones among them, have tried various ways to attack this problem, and it is still as elusive as it was when first posed. So the importance of the problem is that ...

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Is there any way to define arithmetical multiplication as other thing than repeated addition?
39 votes

Given two sets $A$ and $B$ of cardinality $a$ and $b$, respectively, the cardinality of the cartesian product $A\times B$ is called the product of $a$ and $b$, and is denoted by $a\cdot b$. Update ...

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Nice expression for minimum of three variables?
38 votes

Here is a hint why there is no simple such formula: In the case of two variables $a_i$ they are the zeros of the polynomial $$p(x):=(x-a_1)(x-a_2)=x^2 -(a_1+a_2)x + a_1 a_2\ .$$ Therefore by the ...

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Appropriate Notation: $\equiv$ versus $:=$
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35 votes

An "equality by definition" is a directed mental operation, so it is nonsymmetric to begin with. It's only natural to express such an equality by a nonsymmetric symbol such as $:=\, .\ $ Seeing a ...

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Why do units (from physics) behave like numbers?
34 votes

Electrical tensions are not numbers but vectors in a one-dimensional vector space of tensions. Choosing the unit "volt" means choosing a basis in that vector space. In this way each tension is then a ...

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Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$
Accepted answer
34 votes

Note that $\sqrt{7}^{\sqrt{8}}\doteq15.673$ and $\sqrt{8}^{\sqrt{7}}\doteq15.656$, so these two are pretty close. Furthermore $\sqrt{7}<e<\sqrt{8}$, and the function $f(x):={\log x\over x}$ has ...

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Is there an elegant bijective proof of $\binom{15}{5}=\binom{14}{6}$?
33 votes

Assume that we have to choose from $15$ people a committee of $5$ plus a president and a secretary. (a) We can first choose the ordinary members in ${15\choose 5}$ ways, then the president in $10$ ...

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The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?
33 votes

A hint: Think about the remainder modulo $9$.

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$6$ women and $4$ men wait in line. If their order in line is random, find the probability that all of the women are adjacent to one another.
Accepted answer
32 votes

Some hints: $10$ people can be arranged in a line in $10!$ ways. Four men and a bench can be arranged in $5!$ ways. Six women can be placed on the bench in $6!$ ways.

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Mean Value Theorem for complex functions?
Accepted answer
31 votes

As mrf has shown there is no general inequality of the conjectured kind. But you can argue as follows: $$\bigl|\phi(z)-\phi(z_0)\bigr|=\left| \int_\gamma \phi'(\zeta)\ d\zeta\right| \leq \int_\gamma \...

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Why is the Möbius strip not orientable?
31 votes

Let $M:=\{(x,y)|x\in\mathbb R, -1<y<1\}$ be an infinite strip and choose an $L>0$. The equivalence relation $(x+L,-y)\sim(x,y)$ defines a Möbius strip $\hat M$. Let $\pi: M \to \hat M$ be ...

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Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.
31 votes

The function $$f(x):=\cos(\sin x)-\sin(\cos x)$$ is even and $2\pi$-periodic; therefore it suffices to consider $x\in[0,\pi]$. When $x=0$ or $x\in\bigl[{\pi\over2},\pi\bigr]$ then obviously $f(x)>0$...

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Why does a distance and its square reach their minimum at the same point?
30 votes

This has nothing to do with derivatives, nor epsilons; it's pure logic. If you have a function $f:\>P\to{\mathbb R}_{\geq0}$ defined on some set $P$ (like a parabola in the plane) and a strictly ...

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About interesting numbers
29 votes

Interesting numbers can only have prime factors $<30$. There are $10$ such primes. For the purposes of this problem for each interesting number $n$ and each prime $<30$ it is only important ...

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Recovering a quadratic polynomial from three values using calculus
29 votes

This problem has nothing to do with calculus. Knowing about the symmetries of the quadratic function one can proceed as follows: Make $(2,7)$ your origin. This amounts to introducing the function $$g(...

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