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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Condition for a function to change its sign at $x=a$
Consider a situation in which we are only allowed to evaluate the value of a real function $f$ (and its derivatives) at a particular value $a$. Also, assume that the given function is $C^\infty$ ...
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Example for application of theorem: $G = \langle C_G(a) \mid a \in Q \setminus \{1\}\rangle$
Let $p, q$ be distinct prime numbers, $G$ a $p$-group, and $Q$ a non-cyclic abelian $q$-group of automorphisms of $G$. Then,
$$ G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle . $$
Hey guys,
...
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Is $e^x \cos(e^x) f(x)$ absolutely integrable on $\mathbb{R}$ for any Schwartz function $f$?
In this ME post, it is said that the function $e^x \cos(e^x)$ is a tempered distribution on $\mathbb{R}$. Namely, we have
\begin{equation}
\int_{\mathbb{R}} e^x \cos(e^x) f(x) dx = -\int_{\mathbb{R}} \...
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Show that the $\sigma$-finiteness assumption of $\mu$ cannot be omitted in Radon-Nikodym Theorem
The Radon-Nikodym Theorem says the following:
Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, and let $\mu$ and $\nu$ be $\sigma$-finite positive measures on $(X,\mathscr{A})$. If $\nu$ ...
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Prove carefully $C^1[0,1]$ is incomplete
This post shows how to prove $C^1 [0, 1]$ is incomplete in the uniform norm. But I want to get a deeper understanding, specifically how to come up with an example. Here's my understanding:
I know $C^0[...
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Application of theorem: Group with fixpointfree automorphism of order 2 is abelian.
Let $G$ be a group and $a \in Aut(G)$ with $o(a)=2.$ If $C_G(a)=1$, then $x^a=x^{-1}$ for all $x \in G$. In particular, $G$ is abelian.
Hello, does anyone have an example where this theorem can be ...
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Relative homotopy equivalence despite not being a retract?
The following chain of implications, for a subspace $A$ of $X$, is well-known and easy:
($(X, A)\cong (Y, B)$ means that there exist $f\colon X\to Y$ and $g\colon Y\to X$ such that $f(A)\subseteq B$, ...
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Find an example of connected topology for finite space $\{1,...,n\}$ that conforms with the intuition of a commoner.
The most common topology for finite space is the discrete topology, which is clearly not connected.
There are some other discussion on this site, which give many example topology of finite space and $\...
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Quantifying how "concentrated" a distribution is
Consider $n\in\mathbb{N}$ buckets with infinite capacity and let $x\in[0,1]^n$ with $\sum_{i=1}^n x_i=1$ be a way to distribute 1 litre of liquid across all $n$ buckets, i.e. $x_i$ is the amount of ...
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Convergence in $L^1_\text{loc}$ of weak derivates
Let $I$ an open inteval in $\mathbb{R}$. I'm looking for an example of a sequence of functions $(u_n)$ and a function $u$ in $I$ such that $u_n$ for all $n$ and $u$ has weak derivates $u_n'$ and $u'$ ...
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Counterexample for a proof
Let $n$ and $k$ be positive integers and
$$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$
be a lattice cube of length $n$.
Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red ...
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On the permutation of vertex set and automorphism of graphs.
I’m novice in graph theory, I greatly appreciate if you find any mistake and edit that mistake.
Suppose I have a unlabelled simple, undirected, graph $G$. Vertex set consist of $N$ vertices. Now, we ...
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Which topologies admit exactly one open singleton and have more than half of the possible subsets open?
I want to find all topologies $\tau$ on finite set with $n$ elements such that all topologies have exactly one open single point or exactly $\{a_0 \}$ be in topology and cardinal of topologies $\tau$ ...
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Determine if inner product over a real vector space has a certain form
Verify if the following statement is true: Every inner product on $\mathbb{R}^n$ has the form $\langle v,u\rangle = v(Au),$ where $A$ is a symmetric matrix with positive entries on the diagonal.
I ...
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Can Cauchy's polyhedron rigidity theorem be generalized to affine transformations?
Conjecture: Suppose $f$ and $g$ are two convex realizations of an abstract polyhedron $P$. (In other words, $f(P)$ and $g(P)$ are two convex polyhedra whose face lattices are isomorphic.) Also suppose,...
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