The counterexamples tag has no wiki summary.

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### Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing.
The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of ...

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168 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 ...

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### Bialgebras with Hopf restricted duals

It is known from the general theory that, given a bialgebra (over a field $k$)
\begin{equation}
\mathcal{B}=(B,\mu,1_B,\Delta,\epsilon)
\end{equation}
the Sweedler's dual $\mathcal{B}^0$ (called also ...

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103 views

### Spaces for which separable is equivalent to second-countable

While it is well known for metric spaces, being separable is equivalent to be second-countable. In this post I give a counterexample for a non metric space.
What are other topological properties that ...

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346 views

### Counterexample for associativity of smash product

In Section 1.7 of Parametrized Homotopy Theory by May and Sigurdsson it is stated that the smash product of pointed topological spaces is not associative (which is just another hint that ...

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104 views

### Is a concretely reflective full concrete subcategory necessarily finally dense?

On p.371 of "The Joy of Cats", by J.Adamek H.Herrlich and G.E.Strecker:
Proposition 21.32
If a topological category $(\mathbf{A},U)$ is a finally dense full concrete subcategory of $(\mathbf{B},V)$, ...

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172 views

### Colimit density and monads

Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?

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220 views

### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...

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221 views

### counterexample regarding quotient algebras in forcing

Suppose $A$ and $B$ are complete subalgebras of a complete boolean algebra $C$. Let $G \subseteq A$ be generic. In the extension $V[G]$, we can define the quotient algebras $B/G$ and $C/G$ in the ...

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945 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 ...

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327 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 $\sum_{n = 1}^{\infty}c_nf_n$ ...

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109 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 ...

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### 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 ...

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194 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 $(x\wedge y)(x\vee y)=xy$?

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262 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 $|\frak T|\le |\Bbb R|$?

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### 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 ...

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656 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 ...

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### 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 ...

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### 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 ...

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207 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 ...

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### 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
...

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**1**answer

148 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 $(X,\le)$?

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### 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 ...

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### 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 ...

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152 views

### Does the going-up theorem hold between flat algebras?

Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?

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208 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 $M$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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534 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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335 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 ...

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### 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:
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...