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.
5,879 questions
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Can there be a sequence of invertible matrices over natural numbers?
I am trying to find an example of $A_n$ be sequence of $n\times n$ invertible matrices over $\mathbb{N}$(i.e takes natural numbers as entries), and $A_{n+1}(i,j)=A_{n}(i, j) \ \forall 1\le i,j\le n$ (...
- 9,505
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When does the inverse for a sequence of matrices exist?
Let $A_n$ be sequence of $n\times n$ invertible matrices over $\mathbb{C}$, and $A_{n+1}(i,j)=A_{n}(i, j) \ \forall 1\le i,j\le n$ (i.e we obtain $A_{n+1}$ from $A_n$ by adding a new row and column at ...
- 9,505
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Fekete's lemma for Banach lattice sequences
Recall that a real sequence $(a_n)$ is sub-additive if
$$\forall n, m \in\mathbb{N}^*, ~ a_{n + m} \leq a_n + a_m$$
Fekete's lemma states that if $(a_n)_{n\in\mathbb{N}^*} \in \mathbb{R}^{\mathbb{N}^*}...
- 21.3k
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About diagonal set
Let X be a compact Hausdorff space, and let
$\Delta = \{ (x, x) \in X \times X \mid x \in X \}$
be the diagonal in $ X \times X$ .
Consider the following two statements:
1.$X $is metrizable.
2.$ \...
- 4,784
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Explain this seeming contradiction in Euclid Book 1 Proposition 16
This proposition has been concluded without the use of the parallel postulate, because the first time Euclid invokes the parallel postulate is in I.27. Thus, it should apply to all geometries ...
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Approximating sequences of simple r.v.'s by martingales
Let $(X,\mathcal B,P)$ be the probability space with $X=[0,1]$, $\mathcal B$ the Borel $\sigma$-algebra and $P$ the Lebesgue measure.
Consider an arbitrary sequence $(X_n)_{n\ge1}$ of simple random ...
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Algebraic sum of two semi-continuous functions
Definition. Let $X$ be a topological space and let $A\subseteq X$. We say that $A$ is semi-open if $\overline{A^\circ} \supseteq A$.
Definition. Let $X$ and $Y$ be topological spaces and let $f: X\to ...
- 10k
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Conjecture: a constant-free equation is solvable in a group $G$ if and only if it is solvable in its generating set $B$ for any $G=\langle B\rangle$
Conjecture.
Let $G$ be a group and $B$ any set of generators for $G$. That is to say $G = (G, \cdot) = \langle B \rangle$. Then for any equation $E=F$ in $G$ that is constant-free, we have that $E=...
- 22.8k
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If $H$ and $N$ are respectively a pronormal and normal subgroup then is $H\cap N$ pronormal in $N$?
If $H$ is a subgroup of a group $(G,\ast,e)$ then it is said pronormal iff for any $g$ in $G$ there exists $x$ in $\left\langle H\cup(g\ast H\ast g^{-1})\right\rangle$ such that the equality
$$
g\ast ...
- 5,410
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If $H$, $K$ and $L$ are subgroup such that $H$ commutes with $K$ and $L$ then is $H\ast K\ast L$ a subgroup?
If $H$ and $K$ are subgroup of a group $(G,\ast,e)$ then I know that $H\ast K$ is a subgroup of when $H$ is commutable with $K$: so I am searching a counterexample showing that if a subgroup $X$ ...
- 5,410
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Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
- 199
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Counter example to dominated convergence for nets
I came across the following counter example in the accepted answer to this questions A net version of dominated convergence?
For context, the original example is this:
Let $\Lambda$ be the set of ...
- 3,357
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Counterexample for Unitary Operators and Orthogonal Complements
I'm working on an exercise of Friedberg's linear algebra book. In the previous parts of the exercise, I proved that if $U$ is a unitary linear operator on an inner product space $V$, and $W$ is a ...
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Can there be a continuous function with infinite derivative everywhere?
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that
$$\frac{f(...
- 9,505
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Function continuous nowhere whose domain and range are $[0,1]$
Find a function continuous nowhere, whose domain and range are both $[0,1]$.
My intuition was to start with $f(x)=x$ and exchange to $f(a)=b$, $f(b)=a$ for sufficiently many pairs of $(a,b)$. So I ...
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