# 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.

4,570 questions
Filter by
Sorted by
Tagged with
0answers
11 views

### What is the turing degree of different concrete problems? (looking for examples)

Despite there is a lot of information about the lattice of turing degrees, I have found very little information on the turing degree of a known uncomputable object, I'm looking for examples. This is ...
3answers
192 views

### How many primes of the form $3^n + n^3$ exist?

I was looking for primes of the form $3^n + n^3$ or equivalently of the form $9^m + 8 m^3$. I was not able to find any ?? How many exist ? What are the first few ? It seems like a trivial thing I ...
0answers
25 views

### Example that $\dim(A/P) = \infty$ [duplicate]

If $A$ is a noetherian ring and $P\subset A$ is prime ideal then is there any example such that $\dim (A/P)=\infty$? Here, $\dim$ is a Krull dimension.
1answer
59 views

### Examples of interesting cones in infinite-dimensional Hilbert spaces

Let $X$ be a real Hilbert space, and let $K$ be a closed convex cone. The dual cone is defined by $K^*=\{x^*\in X \mid (\forall k\in K)\, \langle x^*,k\rangle \geq 0 \}$. I am looking for some ...
0answers
31 views

### Example of nilpotent profinite group

Is there any example of a topologically finitely generated profinite group $G$ which is nilpotent as a group but has an infinite torsion-part? The question is motivated by the fact that an algebraic ...
1answer
55 views

### Graph theory:Combinatorics [closed]

(a) Find a graph with degree sequence 4, 4, 3, 3, 2, 2, 2, 2. (b) Show that any graph with this degree sequence is planar. (c) Is it true in general that if a graph is planar, then every graph with ...
1answer
101 views

1answer
14 views

### Properties of $A$-inclusion topology.

Let $X$ be a non-empty set and $A\subset X$.Define $\tau$ on $X$ as follows: $U$ is open iff $A\subset U$ or $U=\phi$.We can easily verify that this is a topology on $X$.Now what are the ...
1answer
66 views

### Group actions on product varieties by diagonal

Let $X,Y$ be two varieties, and $G$ be a group together with actions on $X$ and $Y$. Moreover, we assume the action on $X$ is free. I would like to know if the following statement is true: There is ...
1answer
51 views

### Example of Riemannian metric on sphere such that it become non strictly convex.

My understanding of strictly convexity of the compact set in Euclidean space is that if we take any straight line joining any two boundary points then the line must be in a compact set with out ...
2answers
38 views

### Prove that the following piecewise function is Borel measurable

This question is a follow-up to this question. I am trying to prove that $f(x) = \begin{cases} \dfrac{1}{b}\; \text{ if$x$is rational and$x= \dfrac{a}{b}\$ is in its lowest terms}\\ ...