# Questions tagged [counterexamples]

A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.

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### Examples and counterexamples to Lack's coherence observation

In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...

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### A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited

Can we find a counterexample to the following assertion?
Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto ...

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### Existence of nontrivial categories in which every object is atomic

An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...

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### Separable metrizable spaces far from being completely metrizable

I came across a kind of separable metrizable space that is "far" from being completely metrizable. Before specifying what I mean with "far", I recall that a space is said to be ...

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### Failure of Baire's grand theorem when the hypothesis is weakened to separable metric space

The statement of Baire grand theorem gives a characterization of Baire class 1 functions between a completely metrizable separable space (aka Polish space) and a separable metrizable space. The ...

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### Odd partition with extra properties

Can such a set $A=$ {$a_1,.. a_k$} exist, such that:
$\sum_i a_i = 1$ and $a_i $ are rational positive numbers
$k$ is and odd number, and is at least $3$.
We can partition $A$ in two parts of value $...

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### $\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$

I am a PhD student and during my research I was presented to the claim that
For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, ...

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### Prove or disprove the positivity of the ess inf of a singular function

Consider a measurable radial function $u:\Bbb R^d\to(0,\infty)$ such that
$$\int_{B_\delta(0)} u(x) d x=\infty\quad\forall\,\, \delta>0.$$
I would like to prove or to disprove that there exists $r&...

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1
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### Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$

A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' Classical Descriptive Set Theory, p. 192):
Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\...

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### Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...

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### Mañé's example of an attractor with no natural measure

I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...

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### Hoeffing inequality is not true for stopping time

Let $X_k$ be a sequence of iid Bernoulli random variables of parameter $p$ and let $\hat{X}_n=\frac1n\sum_{k=1}^nX_k$. Hoeffding's inequality states that for any $n$:
$$\mathbb{P}(\hat{X}_n - p \ge \...

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### Find a Borel measure such that the closed sets aren't arbitrarily close to the Borel sets with finite measure

I would like example of measures which shows that the following propositions are false:
Proposition 1: Let $\mathfrak{B}$ be the Borel $\sigma$-algebra of a topological space $X$ and $\mu:\mathfrak{B}...

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### What is wrong with the experimental evidence against the semi strong perfect graph theorem?

We got experimental evidence against the semi strong perfect graph theorem
and would like to learn what is wrong with it.
From Recognizing the P4-structure of bipartite graph
The P4-structure of a ...

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2
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### Hausdorff quasi-Polish spaces

A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. ...

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### $\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$

I'm hoping to find an explicit construction for a sequence such that $\sum a_n = 0$ but $\sum \frac{a_n}{n} = \infty$, or a proof that one cannot exist. So far, I have a good idea of how we can ...

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### Examples of 2-categories with multiple interesting proarrow equipment structures

Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ...

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### Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...

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### Are there infinite abelian real reflection groups?

Lately I have been studying reflection groups, and there is a particular example of a complex reflection group that has been very good for guiding my intuition. I would like to know if there is an ...

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1
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### Lower bounds on translates of a function over a compact set

Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. Let $K\subset\mathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'\...

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1
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### Lower bounds on translates of a function

Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'}\frac{\Vert f_\theta - f_{\theta'}\Vert_p}{|\...

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1
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### Good prime ideals in tensor products of local rings

Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...

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### Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?

I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself.
Say we are working on the unit disc $D \subset \mathbf{R}^n$, ...

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3
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### Hausdorff convergence in bounded set preserves the volume

I was wondering if Hausdorff convergence relates to the volume of the converging sets. In particular, let $(C_n)$ be a sequence of closed sets contained in a bounded, closed set $Q$. Assume that $|C_n|...

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1
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### Examples of $C^{k,1}$ functions which are not $C^{k+1}$?

I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...

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### Anomalous phenomena [closed]

What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when ...

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### Can a manifold be triangulated with minimal surfaces

It is a fact stated as an exercise in chapter 9 of Lee's book "Riemannian Geometry" that any compact 2D manifold can be triangulated by geodesic triangles. Can one triangulate any compact ...

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### Nonseparable counterexamples in analysis

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...

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### Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...

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1
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### Looking for non-polynomial functions: with the growth condition: $\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)}$

I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have
\begin{align}\label{...

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### Minimum-weight disjoint union of perfect matchings

Is there a counter example or proof for the claim that the lightest edge-disjoint union of a pair of perfect matchings contains the edges of the lightest perfect matching in a finite complete graph ...

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### Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and $f(...

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### Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...

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### Sufficient coordinate-free condition for points being co-spheric

Question:
is there a theorem that guarantees that
$\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-...

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### Dense sub-algebra of $C_{b}((0,1))$ which is not smooth

I am looking for a dense sub-algebra $B$ in $C_{b}((0,1))$ in uniform topology such that it satisfy following requirements:
$B\cap C^{\infty}_{b}((0,1))=\mathbb{R}$ (No polynomial, no bump function).
...

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### Low-Dimensional Spaces with High-Dimensional Homology

Barratt-Milnor Spheres $X_n$ are spaces with finite topological dimension $n$ but which have non-vanishing singular homology in arbitrarily high dimensions. Here, they prove that if $n > 1$ then ...

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1
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### Uniform approximation of indicator function of a point

Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that
$g_n$ converge uniformly ...

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0
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### Sequence of functions tending to zero in L^2

Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...

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### Smallest known counterexamples to Hedetniemi’s conjecture

In 2019, Shitov has shown a counterexample (Ann. Math, 190(2) (2019) pp. 663-667) to Hedetniemi’s conjecture,
$$\chi(G \times H)=\min(\chi(G),\chi(H))$$
where $\chi(G)$ is the chromatic number of the ...

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### Covering image of a connected CW-complex need not be a CW-complex

This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...

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### Is a cohesive set always an almost subset of a co-simple set?

A set $A\subseteq\omega$ is called a cohesive set if $C$ is finite for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite. And a set $A\...

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### A "proof" that all separately continuous maps on LF-spaces are continuous

Problem
Consider the locally convex spaces $C^\infty(\mathbb{R})$ and $C^\infty_c(\mathbb{R})$, the former equipped with its standard Fréchet topology, the latter equipped with the inductive limit ...

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### Estimator preferred over the other [closed]

Suppose $\theta_1$ and $\theta_2$ two estimators of the mean $\mu$ knowing that $MSE(\theta_2) = MSE(\theta_1)$, $\theta_1$ estimates $\mu$ with a bias and $\theta_2$ estimates $\mu$ without a bias. ...

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### Which domains have a Poincare-Wirtinger inequality? Which don't?

A Poincare-Wirtinger inequality holds over a domain $\Omega \subseteq \mathbb R^n$ with exponentnt $1 \leq p \leq \infty$ if there exists $C(p,\Omega) > 0$ such that
$\| u - \operatorname{avg}(u) \|...

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1
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### Primality test for numbers of the form $4k+3$

Can you prove or disprove the following claim:
Let $n$ be a natural number of the form $4k+3$ , and let $c$ be the smallest odd prime such that $\binom{c}{n}=-1$ , where $\binom{}{}$ denotes a Jacobi ...

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### Isotopy Classes of Non-Connected Planar Sets

I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and ...

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### A weakly sequentially continuous operator which is not weakly continuous

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...

2
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0
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### Failure of Strichartz estimates for the wave equation: elaboration of a counter-example

One can read in Oh - Probabilistic perspectives in nonlinear dispersive PDEs (Proposition 64, p. 60) that there exists a function $F \in L^2_tL^{1}_x (\mathbb{R}_t\times \mathbb{R}^3_x)$ which is ...

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### Primality testing using Chebyshev polynomials

Can you provide a proof or a counterexample for the claim given below?
Inspired by an alternative definition of the Frobenius primality test which is given in this paper I have formulated the ...

5
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1
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### Infinitely many counterexamples to Nash-Williams's conjecture about hamiltonicity?

Question from 2013
gives one counterexample to Nash-Williams's conjecture about hamiltonicity
of dense digraphs.
Later, we found tens of counterexamples on more than 30 vertices
and believe there are ...