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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What is the turing degree of different concrete problems? (looking for examples)

Despite there is a lot of information about the lattice of turing degrees, I have found very little information on the turing degree of a known uncomputable object, I'm looking for examples. This is ...
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3answers
192 views

How many primes of the form $3^n + n^3 $ exist?

I was looking for primes of the form $3^n + n^3$ or equivalently of the form $9^m + 8 m^3$. I was not able to find any ?? How many exist ? What are the first few ? It seems like a trivial thing I ...
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0answers
25 views

Example that $\dim(A/P) = \infty$ [duplicate]

If $A$ is a noetherian ring and $P\subset A$ is prime ideal then is there any example such that $\dim (A/P)=\infty$? Here, $\dim $ is a Krull dimension.
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1answer
59 views

Examples of interesting cones in infinite-dimensional Hilbert spaces

Let $X$ be a real Hilbert space, and let $K$ be a closed convex cone. The dual cone is defined by $K^*=\{x^*\in X \mid (\forall k\in K)\, \langle x^*,k\rangle \geq 0 \}$. I am looking for some ...
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31 views

Example of nilpotent profinite group

Is there any example of a topologically finitely generated profinite group $G$ which is nilpotent as a group but has an infinite torsion-part? The question is motivated by the fact that an algebraic ...
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1answer
55 views

Graph theory:Combinatorics [closed]

(a) Find a graph with degree sequence 4, 4, 3, 3, 2, 2, 2, 2. (b) Show that any graph with this degree sequence is planar. (c) Is it true in general that if a graph is planar, then every graph with ...
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1answer
101 views

Can there exist a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim\limits_{x \to 0}f'(x) = + \infty$.

I think that the following proposition is true: There exists a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(0) = 0$ and for which there exists a sequence $(x_n)_n$ in $\...
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3answers
94 views

If a proposition has only intractably large counterexamples, is it effectively true? [closed]

Consider something like Goldbach's Conjecture. Suppose we determine that it's semi-decidable (I think that's the term?), such that we can't prove it either way short of computing for an unknown amount ...
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0answers
40 views

Second-countable, analytic, completely Baire spaces

First a few definitions. A topological space $(X,\tau)$ is said to be: a Polish space if it is separable and completely metrizable. Analytic, if there exists a surjective continuous map from a (Borel ...
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anyone can help me show the application (maybe an example) of the theorem 3.1 in this journal [closed]

anyone can help me show the application (maybe an example) of the theorem 3.1 in this journal: Theorem 13.1: Let $A:C(\Omega) \to C(\Omega)$ be a compact operator and the equation $$ (I - A)\varphi =...
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1answer
108 views

Find a relation on $\{1, 2, 3\}$ [duplicate]

For each of the eight subsets of {reflexive, symmetric, transitive}, find a relation on $\{1, 2, 3\}$ that has the properties in that subset, but not the properties that are not in the subset. What I ...
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1answer
52 views

If a real sequence $y_{n} = 3x_{n+1}-2x_{n}$ converges, does it imply $x_{n}$ converges? [duplicate]

I'm aware that in the case $y_{n} = x_{n+1}-x_{n}$ this is certainly not true (take for instance $x_{n} = \sqrt{n}$), however I wasn't able to find such a counterexample as easily in this case (if one ...
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1answer
121 views

Artinian, but not Noetherian module: a wonderful example.

I found this wonderful example here. I did not understand some details, could you help me understand? The questions will be asked within the example. Let $p\in\mathbb{N}$ be a prime number. Consider ...
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0answers
53 views

Find a counterexample of: If $1_{A}+1_{B}$ is a random variable then $A$ and $B$ are measurable

I was trying to prove the following proposition, if $1_{A}+1_{B}$ is a random variable then $A$ and $B$ are measurable. Where $1_{A}$ is the indicator, function given by \begin{equation} 1_{A}(x) = \...
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33 views

Blob equation in $\Bbb{R}^3$

I'll start by apologizing for the nature of my request. I'm not sure if this kind of post fits exactly what is expected of the forum, but at the same time I can't think of a more suitable place. I'm ...
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1answer
16 views

If $\sum_{n=0}^{N}a_nx^n-C_N|x|^N\le f(x)\le \sum_{n=0}^{N}a_nx^n+C_N|x|^N$ If $\sup_{n\in \mathbb{N}} C_n<\infty$ then $f$ is analytic

Let $f$ be a function s.t. there exist two sequences $a_n$ and $C_n$ s.t. $$\sum_{n=0}^N a_nx^n-C_N|x|^N\le f(x)\le \sum_{n=0}^N a_nx^n+C_N|x|^N$$ If $\sup_{n\in \mathbb{N}} C_n<\infty$, then $f$ ...
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1answer
266 views

Can every continuous function be ''transformed'' into a differentiable function?

Can every continuous function $f(x)$ from $\mathbb{R}\to \mathbb{R}$ be transformed into a differentiable function , That is,Is there always a continuous (non constant) $g(x)$ such that $g(f(x))$ is ...
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0answers
40 views

Example of a closed and convex but not bounded set in $\mathbb{R}$

Let us consider $\mathbb{R}$. I am trying to find (if exists!) an example of a closed and convex set which is not bounded. Could anyone please help me or give a hint? Thank you in advance!
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2answers
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Riemann Integrability Questions

For each of the following sentences, we assume $f$ is Riemann integrable on $[a, b]$ and $F$ is the function on $[a, b]$ defined by $$ F(x)=\int_a^x f(t) \, \mathrm{d} t $$ Determine whether the ...
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1answer
60 views

Pardox in the counter-example for a closed and bounded set that is not compact.

The following question has been answered in many threads but I have a doubt about the validity of the conclusion and it almost feels like a paradox to me! I am obviously missing something as this is a ...
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1answer
83 views

Example of Atiyah - Mac Donald about chain condition [closed]

Let $G$ be the subgroup of $\mathbb{Q}/{\mathbb{Z}}$ consisting of all elements whose order is a power of $p$, where $p$ is a fixed prime. Then $G$ has exactly one subgroup $G_n$ of order $p^n$ for ...
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1answer
40 views

Non-canonical examples of divergent sequences that are square summable? [closed]

The canonical example of a divergent sequence that is square summable, $\sum_{n = 1}^\infty a_n$ is finite, is the harmonic sequence: $\sum_{n = 1}^\infty 1/n$. Are there examples of sequences that ...
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1answer
25 views

A pseudocompact space not feebly compact

In this question we consider only $T_2$ spaces. A space is pseudocompact if evey continuous $f:X\to \mathbb{R}$ is bounded; it is feebly compact if every locally finite open cover $\{A_i\}\not\ni\...
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2answers
54 views

Tools to show a function is decreasing

I was asked to prove that the following function is decreasing $f(x)=\left(\left(1-\frac{1}{x}\right)^{x}\right)^{x}$ for every $2<x$ I was wondering what tools do I have except the derivate method?...
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1answer
282 views

Proving the non-existence of a sequence satisfying a set of inequalities

There exist some type of conditions called sequential optimality conditions that usually require some inequalities to be ensured along with it, see short introdution. Based on a conjecture on this ...
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2answers
135 views

Being trivial bundles can be checked on linear equivalent divisors?

Let $X$ be a nice (for example, smooth and projective) variety, and let $D_1 \sim D_2$ be two linearly equivalent (smooth and effective) divisors. I would like to know if the following is true: Let $\...
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3answers
61 views

Extension of a bounded continuous function on $[0,1].$

Let $p$ be a prime number, and $S = [0,1] \cap \left \{\frac {q} {p^n}\ \bigg |\ q \in \Bbb Z, n \in \Bbb Z_{\geq 0} \right \}.$ Assume that $S$ has the subspace topology induced from the inclusion $S ...
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1answer
54 views

disprove: $\forall A,B,C(A\cup B=A\cup C \Rightarrow B=C)$

If I let $A=\emptyset$,$B=C=\{x\}$, do I have a valid counterexample?
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0answers
55 views

Domination of derivatives by an involution

Let $h:(0,1) \to (-\infty,0)$ be a $C^1$ function, with $h'>0$. I am looking for sufficient conditions on $h$ that imply the existence of a $C^1$ decreasing* function $g:(0,1) \to (0,1)$ with $g'&...
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2answers
120 views

Can every increasing negative function be expressed as a product with these properties?

Let $h:(0,1] \to (-\infty,0)$ be a $C^1$ function, with $h'>0$. I am looking for sufficient conditions on $h$ that imply the existence of $C^1$ functions $\lambda:(0,1] \to (-\infty,0)$, $g:(0,1] \...
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0answers
75 views

Example where the associated sheaf does not exist

What is a simple example of a topological space $X$ and a complete category $\mathcal{C}$ such that the inclusion $\mathbf{Sh}(X,\mathcal{C}) \hookrightarrow \mathbf{PSh}(X,\mathcal{C})$ has no left ...
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0answers
41 views

I got mixed up by the use of triple bar vs equals sign in linear congruences. Is it worth asking the professor for a regrade?

On an exam I recently took, I was asked the following question: What are the number of solutions to the equation: $ax = b$ (mod $m$) when gcd($b,m$) = 1 and gcd($a,m$) = $d$ > 1 I know this is ...
2
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1answer
60 views

Properties of a sequence of sets [closed]

Let $(X_n)_{n=1}^\infty$ be a sequence of open and bounded sets in $\mathbb{R}^k$ satisfying $$X_1\supset X_2\supset\cdots\supset X_n\cdots$$ and $$\bigcap_{n=1}^\infty X_n=\emptyset.$$ My question ...
4
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1answer
87 views

Abelian and nonabelian groups with the same composition factors.

I think I've answered the question in writing out my thoughts, but I'll share this nonetheless, not as a solution-verification question, but as an examples-counterexamples question. This is Exercise 3....
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1answer
19 views

Example of a risk measure that is not law-invariant

In some theorems about risk measures, the property of law invariance is required. Let $\mathcal{Z} = \mathcal{L}(\Omega, \mathcal{F}, P)$. A risk measure $\rho\colon \mathcal{Z}\to \mathbb{R}$ is law ...
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0answers
55 views

Is $f'$ continuous in a small interval $[0,\epsilon)$?

Let $f:\mathbb R\to\mathbb R$ be differentiable on $[0,\infty)$. Suppose that $\lim_{x\to0^+}f'(x)$ exists and is finite, and that $f'$ is continuous at $0$. Must there exist some $\epsilon>0$ such ...
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1answer
16 views

$\Omega\subset\mathbb{R}^d$ open and bounded and $\phi \in C_b^{\infty}(\Omega;\mathbb{R})\implies$ $\phi$ has bounded derivative?

Let $\Omega\subset\mathbb{R}^d$ be an open and bounded set. Is it possible that a function $\phi \in C_b^{\infty}(\Omega;\mathbb{R})$ (so $\phi$ is smooth, bounded and defined on a bounded open set) ...
2
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1answer
34 views

Example of an unbounded convergent net in a metric space

Please could someone explain how we can obtain a convergent net in a metric space that is unbounded. I have considered indexing a subset in $\mathbb{R}$ by numbers in $(0,\infty)$ as I thought having ...
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0answers
21 views

Chain rule with only partial differentiability assumed

Let $f:\mathbb R^m\to\mathbb R^n$ and $g:\mathbb R^n\to\mathbb R^p$ be two functions that each possess a complete Jacobian matrix of partial derivatives $J_f$ and $J_g$ respectively, which are well-...
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1answer
57 views

Finding an example of a $C^1$ function satisfying a certain boundary condition + derivative condition

I have a real function $f\in C^1([a,b])$. I wish to find another function, $g\in C^1([a,b])$, which is NOT $f$, which satisfies the following conditions: $g(a)=f(a),g(b)=f(b)$ $g'(a)=f'(a),g'(b)=f'(b)...
5
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3answers
133 views

Examples of rational numbers with large denominators appearing unexpectedly. [duplicate]

I am looking for examples of rational numbers with large denominators that pop up in questions in which there is a-priori no large numbers involved or no obvious reason why the denominator would be ...
0
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1answer
22 views

Composite and one-one functions.

I wanted to know if $g \circ f$ is defined and is one-one then under what conditions will both $g$ and $f$ be one-one? Particularly is it possible if $f:A\to B$ and $g:B\to A$?
2
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1answer
35 views

Finding counterexamples to prove: $f_n \to f$ uniformly on $(0,\infty)$ does not imply $1/f_n \to 1/f$ uniformly on $(0,\infty)$

If $f_n \to f$ uniformly, where $(f_n)$ and $f$ are positive functions on $(0,\infty)$, then is it true that $1/f_n \to 1/f$ uniformly on $(0,\infty)$? Solution: This was shown to be false by using ...
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0answers
26 views

Narrow convergence without convergence of the moments.

I thought that convergence in the Wasserstein metric is equivalent to narrow convergence and the convergence of the moments, i.e. $$W_p(\mu_n,\mu)\to 0\iff \mu_n\rightharpoonup\mu\text{ and }\int |x|^...
15
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2answers
208 views

Lawvere theories with two generic objects

What is the simplest example of a category with finite products and two objects $X,Y$ such that $X$ and $Y$ are not isomorphic, $X$ is isomorphic to $Y^n$ for some $n \in \mathbb{N}$, $Y$ is ...
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1answer
63 views

Seeking an example of continuous function that has no integrable derivative

I am seeking for a function $[0,1]\to \mathbb{R}$ that is continuous and has derivative almost everywhere, but this derivative is not integrable niether on sense of improper integrals. For instance, $...
0
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1answer
14 views

Properties of $A$-inclusion topology.

Let $X$ be a non-empty set and $A\subset X$.Define $\tau$ on $X$ as follows: $U$ is open iff $A\subset U $ or $U=\phi$.We can easily verify that this is a topology on $X$.Now what are the ...
1
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1answer
66 views

Group actions on product varieties by diagonal

Let $X,Y$ be two varieties, and $G$ be a group together with actions on $X$ and $Y$. Moreover, we assume the action on $X$ is free. I would like to know if the following statement is true: There is ...
4
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1answer
51 views

Example of Riemannian metric on sphere such that it become non strictly convex.

My understanding of strictly convexity of the compact set in Euclidean space is that if we take any straight line joining any two boundary points then the line must be in a compact set with out ...
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2answers
38 views

Prove that the following piecewise function is Borel measurable

This question is a follow-up to this question. I am trying to prove that $f(x) = \begin{cases} \dfrac{1}{b}\; \text{ if $x$ is rational and $x= \dfrac{a}{b}$ is in its lowest terms}\\ ...

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