# Questions tagged [examples-counterexamples]

To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.

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### Directional derivative zero, minimum and convexity

Let $f:\mathbb R^n \to \mathbb R$ be convex, let $u\in \mathbb R^n$, $v\in \mathbb R^n\setminus \{0\}$ and assume that the directional derivative of $f$ at $u$ in direction $v$ is $0$. I'm wondering ...
1 vote
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### Bound of the norm of $f^{-1}$ in Wiener algebra

Setup. Wiener algebra $W$ is a set of all functions $f(\zeta) = \sum_{n\in \mathbb{Z}}c_n\zeta^n$ on the unit circle with $\|f\|_W = \sum_{n\in \mathbb{Z}}|c_n| < \infty$. The famous Wiener $1/f$ ...
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### Prove or disprove that $\gcd\left(n!+1,f\left(n\right)\right)=1$

Problem/Conjecture : Let : $$f(x)=\sum_{i=1}^{\infty}\left\lfloor\frac{x^{i}}{i!}\right\rfloor$$ Then it seems we have $n\geq 1$ an integer : $$\gcd\left(n!+1,f\left(n\right)\right)=1$$ This question ...
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### Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators

Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators? My motivation for this question is noticing that ...
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### What are examples of Halmos's claim that a single small concrete special case can capture every instance of a concept of great generality?

Paul Halmos states: It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case. What are examples of ...
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### Axler Linear Algebra Done Right 3.f.26

Here's the problem: Suppose $V$ is finite-dimensional and $\Gamma$ is a subspace of $V'$. Show that $$\Gamma=\{v \in V:\varphi(v)=0 \forall \varphi \in \Gamma\}^0.$$ Note that $V'$ denotes the dual ...
1 vote
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### Example of little $\alpha$ Hölder function that is not $\beta$ Hölder, for any $\beta>\alpha$?

What functions are in the set $$c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$ A single example will do, but the more the merrier. This question was natural to me after writing this ...
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1 vote
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### Any complex lattice is equivalent to a lattice of the form …

Follow-up question to this one. A complex lattice consists of a pair of $\mathbb{R}$-linearly independent vectors of the real-vector space $\mathbb{C}$. We call two lattices $(\lambda_1,\lambda_2)$ ...
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### If $G_1$, $G_2$ and $G_3$ are subgroup of $G$ then does the equality $\big\langle⟨ G_1\cup G_2⟩\cup G_3\big\rangle=⟨ G_1\cup G_2\cup G_3⟩$ holds?

So it is a well knew result that any intersection of subgroup is a subgroup and even it is a well knew result that the union of subgroup is not generally a subgroup. However, if $\mathcal S(G)$ is the ...
1 vote
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### $(X,Z) \sim (Y,Z) \implies X \sim Y$? [duplicate]

Let $X,Y,Z$ be random variables such that the random vectors $(X,Z)$ and $(Y,Z)$ have the same law. Is it true that $X$ and $Y$ have the same law ? I can't find a counterexample to it but I can't see ...
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### Non-trivial (regular) open semigroups of the open unit interval $(0,1)$?
This is a follow-up question to the one I asked here. Namely, are there examples of open (or even regular open) subsets of $(0,1)$ which are multiplicative subsemigroups of $(0,1)$ but are not of the ...