Questions tagged [examples-counterexamples]
To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.
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Prime twins satisfy Hardy-Littlewood analogue : $\pi_2(x+y) \leq \pi_2(x) + \pi_2(y) + 2 $?
Inspired by
Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?
and
https://mathoverflow.net/questions/173670/arguments-for-the-second-hardy-littlewood-conjecture-being-false
I ...
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How to extend the following approximations of LogSumExp function into $N$ variables? (metrics defined in question)
How to extend the following approximations of LogSumExp function into $N$ variables? (metrics defined in question)
Intro
I am trying to understand how a portfolio of Geometric Brownian Motions (GBM) ...
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Is Sorgenfrey plane $\Delta$-normal?
Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
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Prob. 9, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: A closed subspace of a separable space need not be separable
Here is (part of) Prob. 9, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Let $A$ be a closed subspace of $X$. . . . Show by example that if $X$ has a countable dense subset, $A$ ...
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Are there finitely many primes of the form $\frac{2^{p^2}\pm 1}{2^p \pm 1}$?
Consider primes of the form $\dfrac{2^{p^2}+1}{2^p+1}$ where $p$ is itself an odd prime.
I know $\dfrac{2^{49}+1}{2^7+1} = 4363953127297$ is a prime number.
MAIN QUESTION : Is this the last one ?
I ...
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Does pasting lemma hold for uniformly continuous functions?
Let $X$ be a metric space with a finite open cover $U_1,\cdots, U_n$. Let $f:X \to \mathbb{R}$ be a function whose restriction to each $U_i$ is uniformly continuous. Is it always true that $f$ is ...
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What is wrong with this "counterexample" to the Extreme Value Theorem?
The extreme value theorem says that any continuous function on a bounded, closed interval has a maximum and a minimum value. I came up with the following construction of a continuous function on a ...
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Non-trivial examples of inconsistent theories
Is there any example of an inconsistent axiom system where the inconsistency is not trivial?
I’d like an example where you could think at first that this theory is not a trivial one or even that it is ...
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Can this be done? Split Pascal's triangle (without the 1's) into two connected regions of equal sums.
Consider Pascal's triangle with $n$ rows, without the $1$'s, with each number in a regular hexagonal cell.
Here is an example with $n=5$:
image source
Does there exist an $n$ such that we can colour ...
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Commuting matrices that are not polynomials or generalized power sums of each.
Let $A$ and $B$ be $n \times n$ matrices over a field such that they satisfy the commutation relation $AB = BA$. It is easy to see that if $A$ is expressed as a polynomial in $B$, such that
$$A = \...
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Pullbacks of manifolds don't exist [duplicate]
Consider the category $\mathbf{Man}$ of smooth manifolds and smooth maps. I wonder how to prove that it doesn't have pullbacks. Since finite products exist, this is equivalent to showing that it doesn'...
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Does $B \mathbf{On}$ have sequential limits?
Question. Consider any sequence of ordinal numbers $\alpha_0,\alpha_1,\dotsc$. Is there a sequence of ordinal numbers $\pi_0,\pi_1,\dotsc$ such that $\pi_n = \alpha_n + \pi_{n+1}$ and which is ...
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Is there a real function whose graph intersects every circle, no matter how big or small?
Is there a function $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that the graph of $f$ in $\mathbb{R}^2$ intersects every circle $C$ in $\mathbb{R}^2$?
I know there are real functions whose graph is ...
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Any Example(s) to Show the Independence of These Two Contractive Conditions?
Let $(X, d)$ be a metric space, and let $f \colon X \longrightarrow X$ be a mapping.
Then $f$ is said to be of $A$-type if there exists a positive real number $\alpha < 1/2$ such that
$$
d \big( f(...
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Why does $f^* = 0$ on cohomology not imply $f_* = 0$ on homology?
In a previous post, A map which is trivial on homology but not on cohomology?, it is shown that a map can be zero on homology groups $H_n$ (all $n\ge1$) while being non-zero on cohomology $H^n$ ($n\...
