The counterexamples tag has no wiki summary.

**36**

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**2**answers

770 views

### Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?

This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about ...

**3**

votes

**1**answer

183 views

### A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...

**0**

votes

**3**answers

85 views

### On the existence of compactly supported functions whose its Fourier transform satisfies a given condition

My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly ...

**2**

votes

**1**answer

70 views

### Counterexample for the Generalized Associativity Equation

The Generalized Associativity Equation is given by
$$ F(G(x,y),z)=K(x,H(y,z)),$$
where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional ...

**3**

votes

**1**answer

170 views

### Cancellative semigroup on a distributive lattice

Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?

**11**

votes

**1**answer

257 views

### Maximum length of a chain of topologies on $\Bbb R$

Let $\frak T$ be a totally ordered set of topologies on $\Bbb R$.
Is $|\frak T|\le |\Bbb R|$?

**1**

vote

**1**answer

109 views

### edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...

**24**

votes

**3**answers

620 views

### Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...

**10**

votes

**3**answers

584 views

### Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...

**4**

votes

**1**answer

85 views

### Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains

Assume $\kappa$ is uncountable and $\phi$ is an $L_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec_{\infty,\kappa}$. It is well-known that $K$ is ...

**0**

votes

**1**answer

197 views

### Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...

**21**

votes

**1**answer

957 views

### Possible counterexample to a theorem assuming Lang's conjecture

Looks like I found a counterexample to a theorem assuming Lang's conjecture,
but not sure it is correct.
Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture
p. 2
...

**2**

votes

**1**answer

131 views

### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

**1**

vote

**2**answers

145 views

### Hausdorffness inheritance in topological groups

Suppose $\mathcal T$ and $\mathcal S$ are two compatible Hausdorff topologies on a group $G$ and $\mathcal R$ is a maximum compatible topology on $G$ with $\mathcal R \subseteq \mathcal T\cap \mathcal ...

**3**

votes

**1**answer

83 views

### existence of more distributive Boolean lattices

Are 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 ...

**1**

vote

**1**answer

193 views

### Failure of Noether normalization

I was in a class recently where we were trying to roughly count the dimensions of certain spaces of rational maps from algebraic curves into closed subschemes $Z \subseteq \mathbb{A}^n$. One way to ...

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**0**answers

62 views

### hyperspaces and selection principals

Two things bother me for which I haven't found an answer yet:
1.Is anyone familiar with an example of a topological space $X$, in which the hyperspace $2^X$ with the upper Fell topology is ...

**3**

votes

**2**answers

323 views

### Examples of complete distributive lattices that are not Heyting algebras

Here is a short question with a possibly simple and short answer:
I need an example of a complete distributive lattice that is not a Heyting algebra which should be an infinite complete lattice that ...

**29**

votes

**15**answers

2k 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 ...

**1**

vote

**1**answer

95 views

### order of convergence of the conditional entropy (2)

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality ...

**5**

votes

**1**answer

169 views

### Conjecture of Spira on the zeros of $\zeta^\prime(s)$

Let $N(T)$ be the number of complex zeros of $\zeta(s)$ with imaginary part between $0$ and $T$, and let $N_k(T)$ be the analogous counting function for the $k$th derivative $\zeta^{(k)}(s)$. Based ...

**6**

votes

**1**answer

328 views

### Group scheme counterexample

Could someone give me an example of a finite group scheme $G$ (over some base $S$) so that $G$ minus a point is still a group scheme over $S$, but not affine over $S$?
Oort mentions that there are ...

**1**

vote

**0**answers

173 views

### Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $ 0 \le \sigma \le \frac12$

Based on limited numerical evidence, I suspect this conjecture.
Conjecture: Fix $ 0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. ...

**2**

votes

**1**answer

122 views

### Realizability of extensions of a free oriented matroid by an independent set

Question:
I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...

**0**

votes

**1**answer

103 views

### order of convergence of the conditional entropy

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g(X_n)$ then by Fano's inequality ...

**20**

votes

**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

**7**

votes

**0**answers

298 views

### A counterexample to a conjecture of Nash-Williams about hamiltonicity of digraphs?

Maybe I am missing something, but found potential counterexample to a conjecture
of Nash-Williams.
According to HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS
The outdegree and indegree sequences of ...

**0**

votes

**4**answers

511 views

### A question on metrizable space

Q1, Does a metrizable space $X$ with $e(X)=\omega$ (i.e., it has countable extent) which is not lindelof exist?
Q2, Let $X$ be the one point lindefication of a discret space of cardinality ...

**4**

votes

**1**answer

108 views

### Strictness of the inequality relating the Iitaka dimension and algebraic dimension

For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension ...

**1**

vote

**1**answer

560 views

### An open problem on general topology

There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.
Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a ...

**22**

votes

**0**answers

829 views

### Manifolds admitting CW-structure with single n-cell

Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented):
When does $M$ not admit (up to homotopy-type) a CW-structure with a single $n$-cell?
By classification of ...

**5**

votes

**1**answer

225 views

### Failure of the Pointwise Ergodic Theorem

It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences.
The counter-example usually given is the Folner sequence ...

**0**

votes

**2**answers

260 views

### Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian)

Suppose $X_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian).
Does a similar statement hold if the variates are pairwise ...

**5**

votes

**2**answers

358 views

### Is this a counterexample to a conjecture about independent domination in cartesian graph products?

VIZING’S CONJECTURE: A SURVEY AND RECENT RESULTS (2009) by Bostjan Bresar , Paul Dorbec , Wayne Goddard , Bert L. Hartnell , Michael A. Henning , Sandi Klavzar , Douglas F. Rall
p.25:
...

**0**

votes

**1**answer

252 views

### Counterexamples for this algorithm for recognizing lexicographic product of graphs?

Found a possible reduction from recognizing lexicographic product of graphs to 2SAT
(since 2SAT is polynomial, the algorithm is polynomial).
Can't prove completeness of the algorithm and since it is ...

**10**

votes

**1**answer

585 views

### Frobenius splitting of Fano varieties

Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...

**10**

votes

**3**answers

2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

**4**

votes

**2**answers

180 views

### Module in category O not generated by a finite set of HWVs.

For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category $\mathcal O$" under the impression that any module in $\mathcal O$ has a finite ...

**4**

votes

**1**answer

451 views

### A Nisnevich cover which is not Zariski

The Nisnevich topology on $Sch$ is a Grothendieck topology strictly finer than the Zariski topology, and the etale topology is strictly finer than the Nisnevich topology.
Colin McLarty asked me for ...

**4**

votes

**2**answers

753 views

### Example for the Sobolev embedding theorem when p=n.

Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then
$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 ...

**0**

votes

**1**answer

231 views

### The part of an operator as an analytic generator

Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
$Y$ is another Banach space embedded in $X$. $A_Y$, the part of $A$ in $Y$
is defined as the operator with domain
...

**32**

votes

**7**answers

3k views

### Example of a manifold which is not a homogeneous space of any Lie group

Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...

**4**

votes

**2**answers

209 views

### Is there a counterexample to this stronger version of James homotopy splitting?

James proved the homotopy decomposition $\Sigma\Omega\Sigma X\simeq \bigvee_{n=1}^\infty \Sigma X^{\wedge n}$. This is a natural homotopy equivalence for a pointed connected CW complex $X$. Here ...

**1**

vote

**1**answer

302 views

### $G_\delta$-diagonal

Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence
${G_n}$ of ...

**19**

votes

**3**answers

1k views

### Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...

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**0**answers

211 views

### Example of function with a certain behavior.

Let $f: R \rightarrow R$. Consider the following properties:
$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$
$$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$
$(2)$ - There is a ...

**6**

votes

**1**answer

906 views

### Haar measure for large locally compact groups

In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...

**52**

votes

**12**answers

4k 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 ...

**22**

votes

**5**answers

2k views

### Existence of simultaneously normal finite index subgroups

It is well known that if $K$ is a finite index subgroup of a group $H$, then there is a finite index subgroup $N$ of $K$ which is normal in $H$. Indeed, one can observe that there are only finitely ...