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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### Conjectured primality tests for specific classes of $k\cdot b^n \pm 1$

Can you provide proofs or counterexamples for the claims given below?
Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following two claims:
First claim
Let $P_m(x)=2^{-m}\...

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**1**answer

229 views

### Intersection of nested open ball in complete metric spaces is nonempty?

My question is that whether the following statement is true or not.
In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies
$$
\exists \epsilon > ...

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324 views

### Primality test for generalized Fermat numbers

This question is successor of Primality test for specific class of generalized Fermat numbers .
Can you provide a proof or a counterexample for the claim given below?
Inspired by Lucas–Lehmer–Riesel ...

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**1**answer

89 views

### Does regular $G_\delta$ imply normal?

I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By regular $G_\delta$, I mean for any closed set $A$,
there exists a countable ...

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79 views

### What is a relation between energy space and $L^p_s-$Sobolev spaces?

We define energy space
$$E= \left\{ f\in \mathcal{S}'(\mathbb R^d): \|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$
and
Sobolev spaces
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^...

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273 views

### Primality test for specific class of $N=k \cdot b^n-1$

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ .
Can you provide a proof or a counterexample to the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(...

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403 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 $N=k\cdot 2^n+1$...

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### Reducing the Dimension of Rectangular Assignment Matrices

Question:
Given a weighted complete bipartite graph $K_{m,n}(U,V,E)$, $m^2\lt n$, are there any counter examples to the assumption, that the edges of the minimum weight maximal bipartite ...

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### Insights from disproofs after counterexamples have been given

Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ ...

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372 views

### Example of a ring with non-finitely generated unit group?

The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...

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### Conjectured initial values of Inkeri's primality test for Fermat numbers

This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...

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**1**answer

70 views

### Average Edge-cost Optimality of MSTs

Are there counter examples to the conjecture, that in a complete, finite and symmetric weighted graph $G\left(V,E,\omega\right),\ E=\lbrace \lbrace i,j\rbrace\subset V\times V\rbrace,\ \omega:E\ni e\...

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262 views

### A locally presentable locally cartesian closed category that is not a quasitopos

This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...

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200 views

### Example of a locally presentable locally cartesian closed category which is not a topos?

The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...

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89 views

### A regular first countable space of cellularity at most $2^\omega$

Let $X$ be a regular first countable space of cellularity at most $2^\omega$.
Is it true that the cardinality of $X$ is at most $2^\omega$?
A cellular family is a family of pairwise disjoint non-...

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### Is there a $\sigma$-metacompact space which is not metacompact?

Recall that a space $X$ is metaLindelof if every open cover of
$X$ has a point-countable open refinement.
A space $X$ is metacompact if every open cover of
$X$ has a point-finite open refinement....

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### 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 $...

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**1**answer

89 views

### Is there a metacompact, normal, CCC space which is not Lindelof

I am looking for a space as in the title, i.e.,
Is there a metacompact, normal, CCC space which is not Lindelof?
A space is ccc iff any family of pairwise disjoint open sets is at most countable.
...

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**1**answer

70 views

### Is there a calibre $\aleph_1$ Moore space which is not separable

A topological space has calibre $\aleph_1$ if for every uncountable sequence $\langle U_\alpha\mid\alpha\lt\aleph_1\rangle$ of nonempty open sets $U_\alpha\subset X$, there is an uncountable subfamily ...

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### Is there a Hausdorff weakly Lindelof space which is not DCCC?

As we know, every regular weakly Lindelof space is DCCC. Here DCCC denotes discrete countable chain condition, a space $X$ has discrete countable chain
condition if every discrete family of non-empty ...

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343 views

### The $n$-th derivative has $n$ zeros. Can such a function be unbounded?

I asked this on Math.SE some days ago, but without any success. For some application I need a formal definition of bell-shaped function. So I had the following idea:
Definition. A $C^\infty$-...

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52 views

### Subgroup embedding properties paranormality and polynormality

The following are subgroup embedding properties introduced by Bah and Borevich.
Definition 1: A subgroup $H$ of $G$ is said to be paranormal if for each $g \in G$, we have that $H^{\langle H, H^g \...

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### Characterisation of supersolvability of a finite group

Definition 1: A subgroup $H$ of a group $G$ is said to be abnormal in $G$ if for each $g\in G$, we have $g\in \langle H, H^g \rangle$.
Definition 2: A finite group $G$ is called a $B$-group if every $...

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130 views

### A cubic system with two nested limit cycles with opposite orientations(2)

The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...

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### Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]

I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...

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137 views

### A transversal matroid whose dual is not transversal

In Oxley's Matroid Theory, Problem 14.8.5, it states that it is (or at least was in 1992) an open problem to determine when the dual matroid of a transversal matroid is also transversal. I had assumed ...

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### The recognition principle and CGWH spaces

The recognition principle [Boardmann–Vogt, May] states that a grouplike algebra over the little $n$-disks/cube operad is weakly equivalent to an $n$-fold loop space. There are technical hypotheses ...

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393 views

### Example of a non-locally connected continuum

Continuum $=$ compact connected metric space.
Let $X$ be a continuum. $X$ is indecomposable means that every proper subcontinuum of $X$ is nowhere dense in $X$.
It is easy to see that if $X$ is ...

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### Looking for a weakly Lindel\"of Tychonoff Moore non-ccc space

Is there a weakly Lindel\"of Tychonoff Moore non-ccc space?
Note that here ccc denotes the countable chain condition; a space $X$ is called weakly Linde\"of if for any open cover $\mathcal U$ of $X$ ...

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**1**answer

185 views

### “Brownian motion” without assuming continuity of path at origin of state space

This question is inspired partly by this question Any reference on Brownian Motion continuity. In this post, the author asked if the following three axioms can define a Brownian motion without ...

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853 views

### How quickly can the derivative of an everywhere differentiable function change sign?

Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...

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172 views

### A result on spaces with countable pseudocharacter and countable tightness

There is a statement as follows:
If a Hausdorff (regular, Tychonoff) space $X$ has countable pseudocharacter and countable tightness, then the closure of any set $Y\subset X$ of cardinality $\le \...

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**1**answer

174 views

### Algebraic independence in polynomial rings over $\mathbb{Z}_n$

Let $R<S$ be an extension of commutative rings with identities (i.e. $R$ is a subring of $S$). We say $s_1,s_2,...,s_k$ are algebraically independent over $R$ iff there is no polynomial $f\in R[...

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### Pronormaliser of a subgroup

The pronormaliser of a subgroup $H$ in a group $G$, denoted $P_G(H)$, is defined to be the set of elements of $G$ that pronormalise $H$. That is, $$P_G(H) = \{g \in G \; | \; \exists x\in \langle H, ...

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277 views

### Whether $\varphi(E)$ is a Jordan measurable set?

Definition: A set $S \subset \mathbb {R^{n}}$ is Jordan measurable if it is bounded in $\mathbb {R^{n}}$ and its boundary is a set of Lebesgue measure zero.
The following conclusion has been ...

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**1**answer

261 views

### Counterexample to Riesz representation for Hilbert modules

For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.
It is often stated in the literature that this does not ...

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**3**answers

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### A counter example for Sard's theorem in the case C^1

I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there?
$...

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### example of an n-transitive but not infinitely transitive group action on a space

Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...

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609 views

### Undirected Negative Cost Cycle Detection - Can Bellman-Ford Fail?

this question is a follow-up to Detecting Negative Cycles in Undirected Graphs.
When checking publications related to the problem of detecting negative cycles in weighted, undirected graphs (the ...

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### Pullback along a Lax Functor

Consider a 2-category of bicategories, lax functors and transformations (I'm actually using double categories but the result should be the same).
Given a cospan $\mathbb{C} \to \mathbb{E} \leftarrow \...

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### Lifting a local section to a global section along a homomorphism of quasi-coherent sheaves

If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property?
For every quasi-coherent sheaf $M$...

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### p-Group satisfying the minimal condition on abelian subgroups

Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?
Obviously such a group cannot be locally finite.
I've ...

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540 views

### Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?

I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...

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### Cocontinuous product-preserving functor between Grothendieck toposes

What is an example of a functor $$F : \mathcal{C} \to \mathcal{D}$$ between two Grothendieck toposes which preserves colimits and finite products, but is not left exact (i.e., does not preserve ...

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188 views

### An example of a frame homomorphism which does not preserve Heyting implication

A frame is a complete lattice $\langle L,\mathord{\leqslant}\rangle$ which satisfies the following distributivity law:
$$a\wedge\bigvee_{i\in I}b_i=\bigvee_{i\in I}a\wedge b_i\,.$$
A frame ...

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247 views

### maximal tensor product of simple $C^*$algebras is non-simple

Let $A$ and $B$ simple $C^*$-algebras. One can prove that the minimal tensor product $A\otimes _{min}B$ is simple. This is wrong for the maximal tensor product $A\otimes_{max}B$ .
1.Do you know an ...

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### Non-isomorphic rings that are localizations of each other

Do there exist commutative rings $A$ and $B$ and multiplicative subsets $S\subseteq A$, $T\subseteq B$ such that $A\not\simeq B$ but $S^{-1}A \simeq B$ and $T^{-1} B\simeq A$?
This question comes ...

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### Reference or counter-example for Closed Graph Theorem for multivalued maps in general topological spaces

Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature ...

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### Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)

We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...

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### Sequentially continuous but not continuous linear map $(X^*, w^*)$ to $(Y^*,w^*)$

Let $X, Y$ be Banach spaces and let $T : (X^*, w^*) \rightarrow (Y^*,w^*)$ be a linear map. Suppose that $T$ is sequentially continuous. Must $T$ be weak*-to-weak*-continuous ?