# 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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### Bound on matrix norms

Let $A,B$ be two $n\times n$ invertible matrices with complex entries. Assume that $B$ is very close to the identity matrix, i.e., $\|B-I\|\ll1$. Can there be a bound on $|\|ABA^{-1}\|-1|$ uniform ...
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### Is this counterexample of $\mathbb{P}( | X | \ge c) \le \frac{\text{Var}[X]}{c^2}$ correct?

Let $X$ be a real valued random variable. Does $\mathbb{P}( | X | \ge c) \le \frac{\text{Var}[X]}{c^2}$ hold for $c > 0$? My counterexample: Let $X \sim \text{Bin}(n,0)$ for $n \in \mathbb{N}$. ...
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### Counter-example wanted on a new formula for convex function

I'm confusing because in the inequality New bounds for convex function of 2 variables the RHS doesn't work (take $f(x)=\tan(x)$) so I propose a new formula that I have checked for all the elementary ...
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### Are there any concrete examples of $\sigma$-algebra generated by a random variable?

I have searched around the internet for any concrete example of $\sigma$ algebra generated by a random variable $X$ but failed to find any nontrivial, concrete examples. For example, https://stats....
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### Function that grows as fast as sum of previous values

Does there exist a strictly positive, monotonically increasing function $f : \mathbb{N} \to \mathbb{R}$ such that $$\lim_{n \to \infty} \sum_{k = 1}^n \frac{f(k)}{f(n)} = 1$$ My guess is that there ...
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### Is a continuous process with finite nonzero quadratic variation a semimartingale?

I know that every semimartingale has finite quadratic variation but the converse is not true. However, what if we assume continuity? If $X$ is a continuous process with finite nonzero quadratic ...
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### When does path connectedness implies convexity?

It is easy to see that every convex set is path connected. What are some examples so that converse holds (not counting the (trivial) one dimensional case)? Is there a nice topology so that this holds? ...
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### Is there any statistically convergent real sequence, which is not almost convergent?

I have read that almost convergence and statistical convergence are incompatible (i.e. not comparable). For this both of below must be satisfied : There exists a statistically convergent real ...
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### $I \cap (J+K) =I \cap J + I\cap K$ [duplicate]

Let I, J, K be three ideals in a commutative ring R with unity. If R is ring of integers then above equation holds. I know the equation do not hold for arbitrary ring. Can you give me an ...
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### Are there any interesting categories/objects whose products are isomorphic to themselves?

This was inspired from an exercise in Lawvere & Schanuel's Conceptual Mathematics. It asks what objects in $\mathbf{Set}$ (finite sets), $\mathbf{S^\circlearrowleft}$ (endomaps in $\mathbf{Set}$),...
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### For ideals $\mathfrak{a,b}$ if a solution $\mathfrak{c}$ to $\mathfrak{cb}=\mathfrak{a}$ exists, is it unique and equal $(\mathfrak{a}:\mathfrak{b})$?

The motivation for this is to have a better intuition of how to think of the ideal quotient $(\mathfrak{a}:\mathfrak{b})$ as a quotient. Obviously for $a,b \in \mathbb{Q}$, the quotient $\frac{a}{b}$ ...
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### Isomorph vector space such that X complete and Y not

Let $X,Y$ be some isomorph vector spaces and let $X$ be a Banach space. If this isomorphism is isometric $Y$ is complete, too. Could someone provide an example such that $Y$ is not complete?
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### A counterexample in measure theory on $\sigma$-infinite spaces

Usually measure theory books include the following theorem (citing Proposition 5.1.3 in Cohn's measure theory book) Let $(X, \mathcal A , \mu )$ and $(Y, \mathcal B, \nu )$ be $\sigma$-finite ...
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### Do we have for all $M \in SL_n(\Bbb K)$, $\lVert M \rVert \geq 1$ when $\lVert \cdot \rVert$ is a matrix norm?

Let be $\lVert \cdot \rVert$ a matrix norm (submultiplicative). Do we have for all matrices of determinant 1, the following lower bound: $$\lVert M \rVert \geq 1$$ I'm very confused and could not ...
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### Is $A$ the $2 × 2$ identity matrix?

If $A$ is a $2 × 2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^2$ have the same characteristic polynomial, then $A$ is the $2 × 2$ identity matrix. My claim: ...
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### Two different versions of limit in topological spaces

Let $X,Y$ be topological spaces, and $f:X\to Y$. We say that $f(x)\to l$ as $x \to b$ iff for every open neighborhood $N$ of $l$, there exists an open neighborhood $M$ of $b$ such that $f(M)\subset N$...
### Compositeness tests for numbers of the form $\frac{k \cdot b^n \pm 1}{2}$
Can you provide proofs or counterexamples for the claims given below? First claim Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $M= \frac{k \cdot b^{n}-1}{2}$ where $k$ ...