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.
278 questions
3 votes
1 answer
434 views
Example of connected, locally connected metric space that isn't path-connected?
I tried to construct a reasonable example for someone, meshing together various sorts of dust, but I either failed or wound up with sets whose separation properties are far too delicate for the ...
- 559
-5 votes
1 answer
235 views
Known examples of conjectures stated while suspected false, to invite counterexamples?
Sometimes, in computational or experimental mathematics, one faces statements that seem almost certainly false yet are not directly refutable by current methods or feasible computation. In such cases, ...
- 1,905
6 votes
3 answers
1k views
Peculiar exception in the number of distinct values taken by the sums of the 6th degree roots of unity
For a nonnegative integer $n$, let $N_n$ be the number of distinct values taken by the sums of $n$ 6th-degree roots of unity (with repetitions). First few counts are $N_0=1$, $N_1=6$, $N_2=19$, and ...
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7 votes
0 answers
150 views
Can image closures of polynomial maps of affine spaces always be surjectively parametrized?
This is a crosspost from my previous math.SE post. Consider an affine variety $X \subseteq \mathbb A^m$ (say, over $\mathbb C$) that is the image closure of a polynomial map $$\phi \colon \mathbb A^n \...
- 375
5 votes
1 answer
274 views
What's the relationship between the Zariski and Scott topologies on the (reverse-ordered) spectrum?
I don't know anything about algebraic geometry. I was bored at work, reading nLab, and noticed that the Zariski topology and Scott topology are vaguely similar. Strictly $T_0$, and almost never ...
- 401
0 votes
2 answers
162 views
continuous, strictly increasing univariate real function with derivative 0 almost everywhere
Are there actually a strictly increasing continuous function from $\mathbb{R}$ to $\mathbb{R}$ with derivative of 0 almost everywhere ? I tried to build one with three real sequences $a_n$, $b_n$ and $...
- 101
6 votes
0 answers
265 views
Around Chertanov's problem
Some time in the 70s Chertanov asked whether there is a compact ccc radial space which is not Fréchet (all spaces are assumed to be Hausdorff). Fréchet means that every point in the closure of a set ...
- 4,925
5 votes
0 answers
421 views
Potential Errors in EGA Chapter $0_{\operatorname{IV}}$ Regarding Formal Smoothness and Formal Etaleness
tl;dr: A collaborator of mine and I have found potential errors regarding formal smoothness and formal etaleness in EGA Chapter 0 and built a potential counter example. Is our counterexample correct ...
- 151
1 vote
0 answers
111 views
Epimorphisms with kernel pairs
I am a bit lost understanding some subtleties in various form of epimorphy. The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one ...
- 261
5 votes
1 answer
274 views
Example: Forgetful functor $\operatorname{CMon}(C)\to \operatorname{Mon}(C)$ is not fully faithful
In our class notes it says that for an $\infty$-category with finite products, the forgetful functor $\operatorname{CMon}(C)\to\operatorname{Mon}(C)$ is in general not fully faithful. Let $C$ be an $\...
- 325
5 votes
1 answer
264 views
Stable infinite category counterpart of pathological behaviours around the AB3,AB4 and AB5 axioms of abelian categories
In his 2002 paper "A counterexample to a 1961 theorem in homological algebra" (Invent. Math.) Amnon Neeman exhibited the infamous and scary example of a cocomplete abelian AB4 (colimits are ...
- 1,008
4 votes
2 answers
257 views
Convergence of the Cesàro mean of iterated continuous functions
Does anyone have a counter-example of the following statement : Let $f : [0;1] \to [0;1]$ a continuous function w.r.t. the usual topology. Let $A_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} f^k(x)$ for $n \ge ...
- 43
1 vote
2 answers
359 views
Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
- 525
7 votes
1 answer
604 views
Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions: $T(X)$ is $s$-closed; $T(X)$ is $...
- 63.4k
2 votes
1 answer
212 views
Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one. I will write everything in detail for the sake of completeness. Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
- 3,745
1 vote
1 answer
185 views
Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
I apologize for repeating the same question from ME, but it seems more subtle than I expected. Let me fix the notations here first: \begin{equation} C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...
- 3,745
4 votes
1 answer
149 views
Inner regularity property of covering number of metric spaces
Let $(X,d)$ be a complete metric space and $n\in\mathbb N$. Suppose that every finite subset $F\subset X$ can be covered by $n$ closed balls of $X$ (that is, $N(Y,d,1)\le n$, in terms of covering ...
- 63.4k
1 vote
0 answers
119 views
When a surjective homogeneous polynomial map is not open at the origin
Question(s): Do there exist surjective homogeneous polynomial maps $f:\mathbb R^2\to \mathbb R^2$ (A) of odd degree less than $5$, and (B) of even degree less than $8$, such that the origin is not ...
- 63.4k
2 votes
3 answers
345 views
Existence of antiderivative w.r.t. any given multi-index for tempered distributions
I originally posted this question on ME, but I find it a lot more nontrivial than expected. So, I post it here. Let $T$ be a tempered distribution on $\mathbb{R}^n$. Then, it is a well-known ...
- 3,745
2 votes
1 answer
378 views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions? I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
0 votes
0 answers
183 views
Question on definition of closed embedding of affine group schemes
$\DeclareMathOperator\Hom{Hom}$I am reading Introduction to affine group schemes by Waterhouse. In the second chapter he defines a closed embedding in the following way. Let $G = \Hom_k(A,-)$ and $H'= ...
- 253
4 votes
1 answer
394 views
Counter example to every closed subscheme $\operatorname{Proj} A$ is of the form $\operatorname{Proj}A/I$
I was under the impression that for a positively graded ring $A$ (not necessarily generated in degree $1$) that every closed subscheme of $\operatorname{Proj}A$ was of the $\operatorname{Proj}A/I$. ...
- 565
1 vote
1 answer
184 views
A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are dense and contractible
Inspired by this question we ask the following question. Note that the comment conversation and answers to the above question imply that There are two complementary subsets of the unit ...
- 226
6 votes
1 answer
311 views
Non-equivalent definitions of Markov process
As far as I know, there are three definitions of Markov processes (or of Markov chains). DEFINITION 1 (WEAKER). A process $(X_t)_{t\in[0,\infty)}$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in ...
- 1,578
2 votes
0 answers
171 views
Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
- 5,009
2 votes
1 answer
160 views
Signed measures on algebras (fields) and their boundedness properties
I asked this question here on math.StackEchange, but it might be too technical so I re-post it here. Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
- 23
1 vote
1 answer
156 views
Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{...
- 1,163
1 vote
0 answers
67 views
counterexample for non- monotone curvature function on the Kazdan-Warner identity
Let $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ be the unit standard sphere, $n\geq 2$. $K(\xi)=\xi_{n+1}+2$, where $\xi=(\xi_1,\ldots,\xi_{n+1})\in \mathbb{S}^n$. It is easy to see that $K(\xi)$ is ...
- 491
1 vote
1 answer
189 views
Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
$ \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\sP}{\mathscr{P}} $ Let $(\Omega, \cA, \mu)$ be a probability space and $\cA_1, \cA_2$ sub $\sigma$-algebras of $\cA$. Let $\...
- 1,163
0 votes
1 answer
154 views
For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?
Sobolev inequalities show us when we can embed a Sobolev space into another. However, I wonder if these inclusions are always proper. More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
- 3,745
4 votes
1 answer
241 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 ...
- 977
5 votes
0 answers
165 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 ...
- 16.1k
6 votes
0 answers
138 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 ...
- 12.5k
7 votes
1 answer
426 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 ...
- 1,097
22 votes
1 answer
1k 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 ...
- 5,039
2 votes
2 answers
310 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 ...
- 13.9k
1 vote
0 answers
41 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
479 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 ...
- 253
3 votes
2 answers
326 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 $C$. (...
12 votes
2 answers
901 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 ...
- 415
7 votes
1 answer
776 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 ...
- 1,329
2 votes
1 answer
238 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. ...
- 1,329
0 votes
2 answers
236 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 ...
- 253
2 votes
1 answer
246 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 ...
- 1,367
2 votes
0 answers
143 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(...
- 121
0 votes
1 answer
142 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 $Df|_x:T_xM\rightarrow T_{f(x)}N$ ...
- 1,329
2 votes
1 answer
272 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/...
- 167
2 votes
2 answers
582 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 ...
5 votes
0 answers
193 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 $\...
- 1,201
5 votes
1 answer
447 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 ...
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