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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.

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Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
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Partial isometries which are shifts

According to this answer: Take copies of Hilbert spaces: $H_1,H_2,H_3,...$ and $K_2,K_3,K_4,...$. Let $A_1$ be a partial isometry which is a shift $H_1 \rightarrow H_2 \rightarrow H_3 \rightarrow ......
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Intuition behind counterexample of Euler's sum of powers conjecture

I was stunned when I first saw the article Counterexample to Euler's conjecture on sums of like powers by L. J. Lander and T. R. Parkin:. How was it possible in 1966 to go through the sheer ...
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1answer
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A special oscillatory orbit in space

Edit: According to the comment of Prof. Eremenko I revise the question. 19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
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1answer
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Banach algebra $A$ without an approximate identity but $A^2=A$

Please help me with the following question. What are some examples of Banach algebra $A$ satisfying the following two conditions? $1$.$ A $ does not have an approximate identity. $2$. $A^2=A$. ...
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1answer
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Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
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Is the projection onto the regular image an epimorphism?

Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$. Let $m:I\hookrightarrow Y$ be the regular image of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ regular mono (i.e. being ...
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An example of a Banach algebra with a specified property

I asked this question (https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions) but unfortunately no one answered it. Please help me to find an ...
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Counterexample in Kolmogorov theorem about existence of almost surely continuous modification

I want to understand this Kolmogorov theorem about existence of almost surely continuous modification: A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...
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Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
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1answer
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If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?

I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
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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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1answer
390 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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1answer
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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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1answer
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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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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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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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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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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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1answer
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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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A complex limit cycle not intersecting the real plane

Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow? There is a regular leaf $L$ whose holonomy, along at least one closed curve ...
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1answer
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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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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1answer
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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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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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1answer
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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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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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1answer
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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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1answer
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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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1answer
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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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1answer
202 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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1answer
889 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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2answers
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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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1answer
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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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1answer
351 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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1answer
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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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3answers
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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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1answer
803 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 \...