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.
248
questions
6
votes
3
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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 ...
5
votes
0
answers
111
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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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votes
9
answers
2k
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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 ...
34
votes
8
answers
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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 ...
3
votes
1
answer
202
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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{...
- 1,033
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votes
1
answer
149
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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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2
votes
0
answers
134
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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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5
votes
1
answer
222
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Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...
- 556
2
votes
0
answers
111
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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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0
votes
1
answer
228
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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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7
votes
0
answers
370
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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 ...
- 409
0
votes
1
answer
339
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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 ...
0
votes
0
answers
154
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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:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
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votes
1
answer
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Smallest known counterexamples to Hedetniemi’s conjecture
In 2019, Shitov has shown a counterexample (Ann. Math, 190(2) (2019) pp. 663-667) to Hedetniemi’s conjecture,
$$\chi(G \times H)=\min(\chi(G),\chi(H))$$
where $\chi(G)$ is the chromatic number of the ...
- 155
14
votes
0
answers
699
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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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votes
1
answer
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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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votes
1
answer
380
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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 ...
0
votes
1
answer
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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. ...
- 101
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votes
0
answers
267
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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) \|...
- 4,776
1
vote
1
answer
352
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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 ...
- 2,683
1
vote
1
answer
79
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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 ...
- 409
1
vote
0
answers
742
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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 ...
- 291
2
votes
0
answers
118
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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 ...
- 489
10
votes
0
answers
626
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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 ...
- 2,683
5
votes
1
answer
269
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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 ...
- 24.2k
2
votes
0
answers
113
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Two more counterexamples to a conjecture from 1975 about hamiltonicity of digraphs
Question from 2013
gives one counterexample to Nash-Williams's conjecture 1975 about hamiltonicity
of dense digraphs.
In the linked answer, @LouisD "reverse engineered" the counterexample
...
- 24.2k
4
votes
1
answer
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When does $\operatorname{Ext}_C^1(M,N_i)=0$ imply $\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0$?
Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\...
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votes
8
answers
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Counterexamples against all odds
What are some examples of conjectures proved to be true generically (i.e. there is a dense $G_{\delta}$ of objects that affirm the conjecture) but are nevertheless false?
Also, it would be cool to see ...
2
votes
0
answers
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if such counter example exists for Lehmer's totient problem could we prove that there are infinity of them or just finitely?
I asked this question one month Ago in MSE but no answer for existence of argument which show if such counter example exists we would have infinity of them or just finitely many examples
Lehmer's ...
- 2,236
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votes
1
answer
372
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A subcontinuous function, which is not continuous
Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be subcontinuous if for any sequences $(u_n)_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u_n)...
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votes
2
answers
525
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fiberwise-quasi-compact implies quasi-compact?
Let $f\colon X\to \mathbb{A}^n_{\mathbb{C}}$ be a morphism of $\mathbb{C}$-schemes. Suppose $f$ is (a) separated, (b) flat, (c) locally of finite type, (d) all fibers are quasi-compact, is $X$ ...
user39380
2
votes
1
answer
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Polish spaces and isomorphisms
An isomorphism between two measurable spaces $(X_1,\mathcal{B}_1), (X_2,\mathcal{B}_2)$ is a measurable bijection $f:X_1\rightarrow X_2$ whose inverse is also measurable.
QUESTION. Can there be an ...
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4
votes
1
answer
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Primality test for specific class of $N=8k \cdot 3^n-1$
This question is related to my previous question.
Can you prove or disprove the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
...
- 2,683
2
votes
1
answer
362
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Primality test for specific class of $N=8kp^n-1$
My following question is related to my question here
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+\...
- 2,683
5
votes
1
answer
328
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Conjectured primality test for specific class of $N=k \cdot 6^n+1$
Can you provide a proof or a counterexample for the claim given below?
Inspired by Theorem 5 in this paper I have formulated the following claim:
Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\operatorname{...
- 2,683
1
vote
0
answers
221
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Recursively obtained hard Diophantine equation for "Baseless numbers"
An equivalent problem was originally asked on MSE as Does every number base have at least one “Baseless number”?, but did not receive any answers that would help answer the main question about "...
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1
vote
1
answer
396
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Convergence in $C_c$ but not in $C$
Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
- 4,969
53
votes
3
answers
5k
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On which regions can Green's theorem not be applied?
In elementary calculus texts, Green's theorem is proved for regions enclosed by piecewise smooth, simple closed curves (and by extension, finite unions of such regions), including regions that are not ...
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votes
1
answer
251
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Is there a finite extension with a non-trivial class group of any PID?
Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?
- 73
11
votes
3
answers
628
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Polynomial inequality of sixth degree
There is the following problem.
Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$
Prove that:
$$\sum_{cyc}\...
- 1,054
1
vote
2
answers
217
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Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...
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1
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Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?
There are loads of concentration results for sums of scalar-valued independent sums $X_1,X_2,\ldots, X_N$ with $\mathbb E[X_n]=0$. For example Hoeffding's Inequality says if all $|X_1|\le 1$ then $\...
- 1,761
1
vote
1
answer
169
views
Non-existent matrices with "essential zeros"
Is there a non-constant continous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and matrices $A=\begin{pmatrix}
a_1 & 0\\
0 & a_2\\
\end{pmatrix}$ and
$B=\begin{pmatrix}
b_1 & 0\\
0 & ...
- 4,969
9
votes
1
answer
415
views
Conjectured primality test for specific class of $N=4kp^n+1$
Can you provide a proof or counterexample for the following claim?
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...
- 2,683
1
vote
1
answer
540
views
Most general form of Jensen's inequality
What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...
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21
votes
1
answer
937
views
Can you give an example of two projective morphisms of schemes whose composition is not projective?
Grothendieck and Dieudonné prove in $EGA_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and if $Z$ is separated and quasi-compact, or if ...
- 46.8k
9
votes
1
answer
802
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Are epimorphic endomorphisms of noetherian commutative rings always injective?
This question was asked, but not answered, on Mathematics Stackexchange.
[In this post "ring" means "commutative ring with one".]
Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism ...
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4
votes
2
answers
244
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Can different knots have the same numbers of quandle colorings for all quandles?
Let $K_1$ and $K_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K_1$ by $X$ is the same as the number of colorings of $K_2$ by $X$. Then my question is, must $K_1$ ...
- 4,529
0
votes
0
answers
23
views
A linear map satisfying the given property
Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying
$T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$.
If the algerba ...
- 167
12
votes
1
answer
1k
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External tensor product of irreducible representations is not irreducible?
I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false.
Dubious claim: If $(\pi, V)$ and $(\rho, W)$ are irreducible representations of two groups $G$...
- 13.1k