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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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Graded endomorphisms factoring through nilpotent ones

Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\...
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2answers
43 views

What is a good example of prefix induction?

the Wikipedia article for Mathematical induction introduces a few variations of the classic principle, such as the strong induction. The strong induction comes with a few examples, namely the closed ...
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2answers
44 views

Counterexample in topological vector space.

Let's $X$ be a topological vector space. So it has two operations $+$ and $\times$. As we know these operations could be continous. But can it be that one of this operation is continous, but the ...
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1answer
28 views

Basic theorem for solvable groups not true for nilpotent groups - counterexample.

it's my first question on MathStackExchange so please be tolerant. Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
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1answer
47 views

Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
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0answers
50 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
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1answer
26 views

Cantor's Intersection Theorem iff complete metric spaces?

If a metric space is complete, we know that the Cantor's Intersection Theorem holds. Does the converse also hold? And if not, what is a suitable counterexample for the same? Also, if the converse ...
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1answer
36 views

Pointwise convergence vs convergence in measure [closed]

I would be very grateful if I could be given an example of a sequence convergent pointwise but it is not convergent in measure and an example of a sequence convergent in measure but it is not ...
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1answer
261 views

“Counterexample” for the Inverse function theorem

In my class we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\Omega$...
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2answers
33 views

Example of $\sum a_n$ convergent but $\sum \sqrt{a_{2^k}}$ divergent?

The problem is this: Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n\geq 0.$ I'm using the Cauchy's method: $\sum \frac{\sqrt{a_n}}{n}$ ...
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1answer
38 views

A hyperbolic but nonstrictly hyperbolic linear conservation law

I need an example of hyperbolic but nonstrictly hyperbolic linear conservation law, but I am not getting one example in any books that I am studying. My main book is LeVeque (1) and the only example ...
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0answers
56 views

Product of two sequential spaces is not sequential

The product of two sequential topological spaces is not necessarily sequential. Could you give an example confirming this? As each first-countable space is sequential and the operation of taking a ...
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1answer
73 views

Inequality $\sum_{cyc}\frac{1}{(7a+b)^7}\leq \frac{\sqrt{3}}{8^7}\Big(\frac{a+b+c}{abc}\Big)^{\frac{7}{2}}$ [closed]

Let $a,b,c>0$ then we have : $$\frac{1}{(7a+b)^7}+\frac{1}{(7b+c)^7}+\frac{1}{(7c+a)^7}\leq \frac{\sqrt{3}}{8^7}\Big(\frac{a+b+c}{abc}\Big)^{\frac{7}{2}}$$ I try many methods like BW,uvw,...
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2answers
283 views

Example in Measure theory about Borel Measures

I need to show that there are distinct $\sigma$-finite measures in $B(\mathbb{R})$ that are the same in the open sets. Im not really seeing a good example cause every open set is a countable union of ...
2
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0answers
33 views

Example of Linear functional on subspace which cannot have extension

I wanted to find example of linear functional on subspace of vector space which cannot be extended to whole vector space. I know that if we are in normed linear space then extension is always ...
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1answer
40 views

Can the real numbers be embedded into all non-Archimedean real closed fields?

Every Archimedean real closed field is isomorphic to a subfield of $\mathbb{R}$. But I’m wondering if something in the opposite direction is true. Suppose that $F$ is a non-Archimedean real closed ...
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2answers
35 views

Example of function $g(x)$ s.t. $E(g|X_n|) \not\to 0$

Consider the following result (assuming probability space ($\Omega, \mathcal F, \mathsf P$)) If $g : [0, \infty) \to [0,\infty)$ bounded, strictly increasing, $g(x) > 0$ for $x > 0$ and $\...
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1answer
24 views

Inequality within complex

I step with a inequality and would like to know if it is truth... $||(a_1-a_2)^2+i(b_1-b_2)^2||\leq ||a_1^2+ib_1^2||+||a_2^2+ib_2^2||,\quad \forall a_1,a_2\in\mathbb{R}$. I tried to prove it but ...
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0answers
9 views

Proposition about field of fractions [duplicate]

A proposition was left in abstract algebra course that is Fields of fractions of different integral domains are different too. I must prove if it’s true and give a counter example if it’s not true....
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1answer
57 views

Example of the Manifold with a Non-Surjective Exponential Map at a Point

I need to find an example of a connected riemannian manifold $(M,g)$ and a point $p \in M$ such that the exponential map $\exp_p : T_pM \to M$ is well-defined, but is not surjective. Taking $\mathbb{...
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1answer
60 views

Example of a bialgebra which is Frobenius but not Hopf

Let $\Bbbk$ be a field. A well-known result (see Larson, Sweedler, An Associative Orthogonal Bilinear Form for Hopf Algebras and Pareigis, When Hopf algebras are Frobenius algebras) states that a ...
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1answer
35 views

Counter-example Aubin-Lions p=1

I wanted to show, that $p>1$ in Aubin-Lions Lemma is necessary. I hoped there already is an example for $X=Y=Z=\mathbb{R}$ so that the embedding \begin{align} \{u \in L^\infty(0,T);\mathbb{R}) \...
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1answer
333 views

Primality test for numbers of the form $N=4 \cdot 3^n+1$

Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer primality test I have formulated the following claim: Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\...
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0answers
35 views

Weakly compact set in $L^1$ not bounded in $L^p$

The Dunford-Pettis theorem states that a family $\mathcal{F}\subset L^1$ is relatively weakly compact if and only if $\mathcal{F}$ is bounded in norm and uniformly integrable, i.e. $\sup_{f\in\mathcal{...
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2answers
47 views

Is $\mathbb{R}^2$ a convex set without extreme points?

My reasoning is, the vector space being a 'flat plane', a straight line joining any two points on it will always be contained in $\mathbb{R}^2$. And it has no points in it which cannot be on the ...
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2answers
41 views

examples where $g$ is a strictly increasing function on an interval say $J$ but $g$ is not continuous.

I am reading Elementary Analysis The theory of Calculus by Kenneth A Ross. Theorem 18.5 Page 130 says Let $g$ be a strictly increasing function on an interval $J$ such that $g(J)$ is an interval $I$...
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1answer
22 views

Does there exist MATRIX games for more than two players

I am in a game theory course and I need to come up with an example of a Matrix game with more than two players I have consulted https://en.wikipedia.org/wiki/List_of_games_in_game_theory, but the ...
2
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1answer
64 views

lim$_{m , n \to \infty} S(n,m)$ exists but iterated limits do not.

$S(n,m)$ is a double sequence. Can anyone give me an example where lim$_{m , n \to \infty} S(n,m)$ exists but lim$_{n \to \infty}$( lim$_{m \to \infty} S(n,m)$) , lim$_{m \to \infty}$( lim$_{n ...
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0answers
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Advantages of Riesz theorem over Caratheodory Extension theorem

I apologize in advance if this question seems vague. I'm self studying Real and Complex Analysis by W. Rudin and I've been reading the proof of Riesz theorem in the second chapter which is then used ...
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1answer
13 views

Not strongly positive operator

Does anyone know an example of a Hilbert space and a bounded linear operator that is positive definite but its induced bilinear form is not strongly positive? Thank you very much in advance.
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1answer
291 views

Counterexample: a pair of linearly ordered sets that are isomorphic to subsets of the other, but not isomorphic between them [duplicate]

I have encountered myself with the following exercise: Let $\langle A, <_R\rangle$ and $\langle B, <_S\rangle$ be two linearly ordered sets so that each one is isomorphic to a subset of the ...
5
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1answer
346 views

Primality test for numbers of the form $N=4 \cdot 3^n-1$

Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer primality test I have formulated the following claim: Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\...
11
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3answers
485 views

Examples of subgroups where it's nontrivial to show closure under multiplication?

Usually when a subgroup is declared, it is trivial (or at least straightforward to a sophomore) to prove that it is a subgroup under multiplication. For example: Homomorphic image and preimage of a ...
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11answers
2k views

Examples of non trivial equivalence relations , I mean equivalence relations without the expression “ same … as” in their definition?

Relations defined by formulas such as " x has the same age as y" , " x comes from the same country as y " " a has the same image under function f as b " are obviously equivalence relations, due to the ...
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2answers
78 views

Partitioning the Reals into two Locally Uncountable, Dense Sets [duplicate]

Is it possible to find two disjoint subsets $X$ and $Y$ of $\mathbb{R}$ such that both are dense in $\mathbb{R}$ and both are locally uncountable? By a locally uncountable set $X \subset \mathbb{R}$, ...
3
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2answers
88 views

Is there always a continuous surjection from $H \times G/H$ to $G$?

Question: Is there always a continuous surjection $f: H \times G/H \rightarrow G$? where $G$ is a topological group, $H$ is a subgroup of $G$ and $G/H$ is given the quotient topology. We know $H ...
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2answers
375 views

How to be good at coming up with counter example in Topology

This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often ...
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4answers
83 views

Examples of contravariant functors

I understand the definition and usefulness of the notion of functor. But I am worrying about the usefulness of the notion of a contravariant functor. Wikipedia writes: There are many constructions ...
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1answer
46 views

A set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point

How can I construct a set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point? I am finding this set in order to ...
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0answers
51 views

Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a random variable $X$, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise ...
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1answer
50 views

Does a topology with countable elements imply that the topology is second countable?

Does a topology with underlying space having countable elements imply that the topology is second countable? Today, the above question come into my mind. This seems very intuitive to me, but I am not ...
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1answer
398 views

Primality test for numbers of the form $N=k \cdot 3^n-1$

Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following claim: Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(...
0
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1answer
28 views

Useful bijections [closed]

Could someone please provide me with some useful bijections one ought to know for an upcoming examination on cardinality with an emphasis on proofs? For example, the bijective mapping $f : (-1, 1) \...
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1answer
89 views

Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
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2answers
61 views

An example of an even numbers $n,k$ whose $gcd$ is three.

I am looking for even integers $n$ and $k$ such that $k$ does not divide $n$ and $\gcd(n,k) = 3$. Is this possible? With the help of some online tools I tried, but every time I am not getting the ...
0
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1answer
43 views

Examples of applying Dirichlet's approximation.

I've seen many examples of Dirichlet's approximation being proven , or other questions regarding to the theory of the approximation on this site and others but I would like to see a concrete example ...
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3answers
450 views

Semigroups with no morphisms between them

Given two monoids we always have a morphism from one to the other thanks to the presence of the identity element. Are there examples of non-empty semigroups that have no morphisms from one to the ...
2
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1answer
68 views

Understanding HNN extensions: intuition, examples, exercises.

What is an HNN extension? What would be some elementary, intuitive examples of them and what exercises involving them would you suggest? The Wikipedia definition is easiest to get to, since neither ...
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1answer
41 views

Counterexample: linearly ordered sets for which there exists more than one isomorphism

In my axiomatic set theory notes, there appears that, if $A$ and $B$ are well-ordered isomorphic sets, then there exists one isomorphism between them. However, as a side note, it is stated that this ...
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2answers
71 views

Counter example for integral differing from limit of integrals

On exercise $4.U$ of Bartle's Element of Integration, we are asked to give a counter example on a sequence $f_n$ of functions such that $\lim_{n\to\infty}f_n=f$, $\int f =\lim_{n\to\infty}\int f_n$ ...