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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3 votes
1 answer
84 views

Are isomorphic maximal tori stably conjugate?

Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
5 votes
0 answers
116 views

If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?

Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
5 votes
0 answers
87 views

Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...
5 votes
1 answer
238 views

Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
7 votes
3 answers
313 views

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. ...
20 votes
1 answer
906 views

Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here. Is the following conjecture true or false: Given any five coplanar points, we can always draw at least ...
2 votes
2 answers
189 views

Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

Question: is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$? I'm convinced it must be true, but can't remember having seen ...
1 vote
0 answers
17 views

Are the categories of definable dinatural transformations freely generated from discrete graphs?

It is well known that the dinatural transformations between multivariant functors defined in Functorial polymorphism don't form a category, because they do not compose in general, but some do. For any ...
7 votes
1 answer
347 views

An example of radical ideal which is irreducible but not prime

$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal. In case $R$ is Noetherian, the radical of $I$ being ...
3 votes
2 answers
271 views

Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
11 votes
2 answers
785 views

Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does ...
7 votes
1 answer
524 views

Composition of power series is power series?

$\DeclareMathOperator\dom{dom}$Sorry to bother the community again with these type of questions about power series, I am ready to delete the question if it is not suitable. Definition: I say a ...
2 votes
1 answer
185 views

Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
0 votes
2 answers
130 views

Examples of isomorphic non-equivalent twisted group algebras

Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
2 votes
1 answer
159 views

Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
2 votes
0 answers
99 views

Real analytic periodic function whose critical points are fully denegerated

I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
0 votes
1 answer
83 views

Must a surjective infinitesmal isometry between simply connected spaces be injective? [duplicate]

Let $f:M\rightarrow N$ be a smooth map between two simply connected Riemannian manifolds of the same dimension. It is also given that for every $x\in M$ we have that $Df|_x:T_xM\rightarrow T_{f(x)}N$ ...
2 votes
1 answer
210 views

Hodge decomposition for non-elliptic complexes

It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
7 votes
2 answers
776 views

Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets (...
2 votes
2 answers
518 views

Can a category be enriched over abelian groups in more than one way?

An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way? An abelian ...
4 votes
1 answer
139 views

Existence of more distributive Boolean lattices

Is there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that $$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in I}a_{if(...
2 votes
1 answer
210 views

A complex limit cycle not intersecting the real plane(2)

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...
5 votes
0 answers
153 views

The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$

Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can $\...
2 votes
1 answer
97 views

Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$

For any positive number $t$, nonnegative integer $n$, and nonzero vector $a \in \mathbb R^n$, define $$ \begin{split} K_n(a,t) &:= \inf_{x \in \mathbb R^n} \|x-a\|_2 + t\|x\|_1,\\ M_n(a,t) &:= ...
5 votes
1 answer
309 views

Simple component that is not a two-sided ideal

Suppose $R$ is a semisimple ring and if $L$ is a minimal left ideal. Let $B$ be the direct sum of all minimal left ideals isomorphic to $L$ ($B$ is called a simple component corresponding to $L$). It ...
4 votes
1 answer
146 views

When does the refinement of a paracompact topology remain paracompact?

Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$. Is it true ...
2 votes
1 answer
157 views

Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
75 votes
13 answers
8k views

Counterexamples in PDE

Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...
127 votes
63 answers
27k views

Counterexamples in algebra?

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and ...
1 vote
0 answers
143 views

What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
3 votes
1 answer
302 views

Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as $$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\...
7 votes
2 answers
530 views

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 ...
13 votes
0 answers
203 views

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 ...
7 votes
2 answers
628 views

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 &...
4 votes
0 answers
128 views

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 ...
2 votes
2 answers
118 views

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 $...
6 votes
0 answers
206 views

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 ...
5 votes
1 answer
222 views

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

$\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\tensor}{\otimes}$ $\newcommand{\colim}{\rm colim}$ $\...
4 votes
2 answers
276 views

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

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 $x$, ...
1 vote
0 answers
69 views

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&...
1 vote
1 answer
116 views

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 $\...
5 votes
1 answer
158 views

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 ...
6 votes
1 answer
156 views

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 ...
1 vote
1 answer
107 views

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 \...
64 votes
3 answers
6k views

Chebyshev polynomials of the first kind and primality testing

Can you provide a proof or a counterexample for the claim given below ? Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim : Let $...
-1 votes
1 answer
72 views

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 ...
37 votes
19 answers
5k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
10 votes
2 answers
901 views

Primality test for specific class of Proth numbers

Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ Let $N=k\cdot 2^n+1$ such ...
1 vote
1 answer
144 views

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}...
2 votes
1 answer
235 views

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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