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 importance of the density of $C_c^\infty(\mathbb R^n)$ and $L_c^\infty(\mathbb R^n)$ on $L^p(\mathbb R^n)$, for every $1 \leq p < \infty$?

Consider the usual Lebesgue spaces $L^p(\mathbb R^n)$, for $1 \leqslant p < \infty$. It is well known that both the spaces $L_c^\infty(\mathbb R^n)$ of essentially bounded functions with compact ...
xyz's user avatar
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What are other graphs of order $n$ than the star $K_{1, n-1}$ which are not packable?

We say, that a graph $G$ is packable, if it is isomorphic to a subgraph of its complement. In more formal terms: A graph $G$ is packable, if there is a permutation $\sigma : V(G) \to V(G)$ such that $$...
anon's user avatar
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2 answers
60 views

$\dim(U_1 \cap U_2 \cap U_3) = n − 3$, Give a proof or find a counterexample.

Suppose that $U_1, U_2, U_3$ are three distinct subspaces of $\dim = n-1$ from a vector space of $\dim = n$. where $n \gt 3$. Give a proof or find a counterexample for $\dim(U_1 \cap U_2 \cap U_3) = n ...
porseshgar's user avatar
0 votes
1 answer
27 views

Attraction of events

I don't know if the next statement is true or false: Let $(\Omega,\mathcal{F},P)$ be a probability space and let $A,B$ and $C$ be events in $\mathcal{F}$ such that $P(A)>0$. If $P(B|A)>P(B)$ and ...
RataMágica's user avatar
4 votes
2 answers
87 views

Example of Complex Pythagorean Triples

I am looking for example of a Pythagorean Triple with Gaussian Integers. I followed the links and looked at followings : Relation to Gaussian integers in https://en.m.wikipedia.org/wiki/...
jimjim's user avatar
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4 votes
3 answers
105 views

If we remove the diagonal from $X\times X$, is it necessarily disconnected?

If $X$ is a compact, connected Hausdorff space, we know that the diagonal $\Delta_X=\{(x,x)\in X\times X\}$ is closed in $X\times X$ by Hausdorffness. But is $X\times X\setminus\Delta_X$ disconnected ...
tripaloski's user avatar
2 votes
2 answers
391 views

Coming up with a counter example - calculus

I have to come up with a counter example for the following statement: Let $f$ be a function $f: [0,\infty)\longrightarrow R$, continuous and bounded. Prove that it receives either a minimum or a ...
natitati's user avatar
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0 answers
94 views

$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .

Conjecture : $$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$ Where $a_n$ is an integer such that : $$0\leq a_n\leq n$$ Some arguments : ...
Miss and Mister cassoulet char's user avatar
5 votes
1 answer
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+200

Building a function $f$ such that $\| f - f_n \|_{L^p(B(x,r) \cap \Omega)} \to 0$ as $n \to \infty$ and $f \in L^p_{\text{loc}}(\Omega)$.

Consider an arbitrary open set $\Omega \subset \mathbb R^n$ and an arbitrary element $1 \leqslant p < \infty$. Moreover, let $(f_n)_{n \in \mathbb N} \subset L^p(B(x,r) \cap \Omega)$ denote a ...
xyz's user avatar
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2 votes
1 answer
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Must an infinite subset of a compact Hausdorff space contain a *sequential* accumulation point?

Def (Sequential Accumulation Point): Given a topological space $(X,\tau)$ and a subset $S\subseteq X$, we say that $a\in X$ is a sequential accumulation point if there exists $(s_n)_{n\geq1}\subseteq ...
tripaloski's user avatar
3 votes
1 answer
75 views

Confused regarding CMI question [closed]

I think there might be an error in the official solution to the following question: CMI May 23, 2022 BSc entrance exam, Question A2(7): You are asked to take three distinct points $1, \omega_1, \...
Soham Saha's user avatar
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1 vote
0 answers
24 views

Counter example for the continuity for decreasing intersections property of measures

Let $\mathscr{R}$ be a $\sigma$-ring and $\mu$ be a (positive) measure on $\mathscr{R}$. Suppose that $\{A_n\}$ is in $\mathscr{R}$ with $A_1 \supseteq A_2 \supseteq A_3 \supseteq\cdots$ and $A = \...
Tran Khanh's user avatar
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1 answer
34 views

Given concave decreasing function, does $\exists c\in [0,1]$ s.t. $\frac{(f(c)-cf'(c))\left(c-\frac{f(c)}{f'(c)}\right)}{2}\leq2\int_{0}^{1}f(x)dx?$

For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ ...
Adam Rubinson's user avatar
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0 answers
15 views

Analysis(convergence of series) If partial sum has a convergent subsequence with lim a_n=0, then series converges?

I have a question. I’m solving a problem; Let <a_n>, <b_n> be two sequence in R. b_k=sum(a_n) from n=2^k-1 to 2(2^k-1) for each k in N. Show that if sigma(a_n)=1, then lim a_n=0 and sigma(...
Brody's user avatar
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0 votes
2 answers
78 views

Real Polynomials on Compact sets of Complex numbers

Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ has domain $\mathbb{C}$. $K$ is a compact subset of $\mathbb{C}$. $\mathbb{R}[x]|_{K}$ is the set ...
John Frank's user avatar
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0 answers
18 views

Second Derivative Test: Can we relax hypothesis that $f$ is twice differentiable on neighborhood? (Counterexample)

Second Derivative Test for Extrema: Let $f:\mathbb R \to \mathbb R$ be a function that is twice-differentiable on $(c-\varepsilon,c+\varepsilon)$ for some $\varepsilon >0$. Suppose $f^{\prime}\...
user182601's user avatar
0 votes
1 answer
98 views

There is a metric space on $\mathbb{Q}$ such that this space be compact.

Prove or disprove: There is a metric space on $\mathbb{Q}$ such that this space be compact. I find some examples of non equivalent metric on rational numbers, euclidean metric, p-adic and discrete ...
amir bahadory's user avatar
2 votes
1 answer
187 views

Let : $\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$ then the minimum over $(0,\infty)$ verify a particular power series

Problem : Let : $$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$ Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have : $$2+A-A^4-30A^6-...
Miss and Mister cassoulet char's user avatar
0 votes
1 answer
66 views

An ordinal $\nu$ is a natural iff there is no injection $f$ of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.

Let's we prove the following theorem. Theorem An ordinal $\nu$ is a natural if and only if there is no injection of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$. Proof. Let's we assume there ...
Antonio Maria Di Mauro's user avatar
0 votes
1 answer
59 views

Finding a Lebesgue integrable function for every $1 \leqslant q < \infty$ that satisfies aditional requirement.

Consider the usual Lebesgue spaces. Amid one of my studies, I started wondering if it is possible to find an example that satisfies the following problem: Problem. Consider arbitrary elements $1 \...
xyz's user avatar
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3 votes
1 answer
56 views

Dirichlet's approximation theorem with even or odd denominators

It follows from Dirichlet's approximation theorem that for any irrational $\alpha,\ 0<\left\lvert \frac{p}{q} - \alpha \right\rvert < \frac{1}{q^2} $ for infinitely many pairs of integers $(p,q)....
Adam Rubinson's user avatar
0 votes
0 answers
49 views

Proof that $P(A \cap B \mid C) \neq P(A \mid C) \, P(A \cap C)$

I want to find a counterexample to verify that the following is false $$ P(A \cap B \mid C) = P(A \mid C) \, P(A \cap C) $$ So I thought I would take $A=C$ and then $$ P(A \cap B \mid C) = \frac{P(C \...
Wrloord's user avatar
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1 vote
1 answer
86 views

Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"

It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
Atom's user avatar
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3 votes
1 answer
125 views

Exercise 1.1.9 from West

Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West. Exercise 1.1.9 states the following: 1.1.9. Suppose that $G$ is a simple graph having no vertex of ...
Bryan Busby's user avatar
3 votes
1 answer
262 views

Can an infinite sum of non-computable numbers be computable, such that all finite sums of subsets of terms are non-computable?

Background In the following question, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers1 indicate the answer to this question is 'yes'. For instance, Rasmus ...
Max Muller's user avatar
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1 vote
0 answers
153 views
+200

Using two copies of a decreasing $k$-tuple, can you form a convex decreasing $k$-tuple greater than the original sequence?

Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k$-tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is, $$ y_n= \begin{cases} x_n&\text{if}\ 1\leq ...
Adam Rubinson's user avatar
3 votes
1 answer
143 views

Why isn't the third property in the definition of vector bundles redundant?

I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows: Suppose $M$ is a topological ...
Kishalay Sarkar's user avatar
0 votes
1 answer
41 views

Example for a particular function

I am looking for an example of a nonnegative function (preferably continuous) that approaches $0$ near $0$, and for any positive number $B$, it is not monotonic on $(0,B)$ and $f(x/2) > f(x)$ for ...
Alejandro's user avatar
  • 107
2 votes
0 answers
31 views

Example of a functor that creates limits but not strictly (other than equivalences)

In Category Theory in Context, Riehl defines creating limits as follows, for a functor $F: \mathbb{C} \to \mathbb{D}$ and diagram $D: I \to \mathbb{C}$: $F$ creates limits if whenever $FD: I \to \...
Anakhand's user avatar
  • 2,532
1 vote
0 answers
33 views

Sequence in $\ell_p$ spaces

The sequence given by: $$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$ That is, $$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$ Is this sequence in the sequence space $\ell_p$ ? where for $...
helen quevedo's user avatar
2 votes
2 answers
150 views

Find three non equivalent metrics on an infinite set.

Find three non equivalent metrics on infinite set. For case infinite countable set we can use this answer for rational numbers. Also for example in $ \mathbb {R}$ we can theree non equivalent metric :...
amir bahadory's user avatar
0 votes
0 answers
52 views

Is there a monotone function that is not differentiable everywhere? [duplicate]

One of the most strange things that I learned in Real Analysis is a function that is continuous on $\mathbb{R}$ but that is not differentiable everywhere. I wonder how to prove that no monotone ...
pie's user avatar
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0 votes
0 answers
75 views

Does every positive real sequence whose series converges to $a$ have a "straight line" convex subsequence that also converges to $a?$

This question is possibly related to my previous question and it's answer. Suppose $a_n>0\ \forall\ n\in\mathbb{N}\ $ and $\displaystyle\sum_{n=1}^{\infty} a_n = a.$ Does there exist a subsequence $...
Adam Rubinson's user avatar
4 votes
2 answers
194 views

Can $x\sin(x)$ be algebraic when it is not $0$?

It's easy to show (using the Lindemann-Weierstrass theorem) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental. But what about $x\sin(x)$? After all, the product of two ...
Ted Hopp's user avatar
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0 votes
1 answer
56 views

Example to show that the containment $\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ is proper where $f$ is continuous mapping

Let $f: X \to Y$ be a continuous function, where $B\subset Y$. Then, $\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ holds, here's the proof. I am looking for an example to illustrate that the above ...
S.S's user avatar
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2 votes
1 answer
72 views

Is there a Hamiltonian cycle of $m$ x $n$ rectangular lattice points (these are the vertices) in $\mathbb{R}^2$ such that no two edges are parallel?

Let $m,n\geq 2$ and consider the rectangular lattice of $mn$ vertices in $\mathbb{R}^2,\ (i,j);\ i\in \{1,2,\ldots,m\},\ j\in \{1,2,\ldots,n\}.\ $ Call these vertices $X_1, X_2, \ldots, X_{mn}.$ Is ...
Adam Rubinson's user avatar
2 votes
1 answer
64 views

Given a convergent sequence of complex numbers, is there a sequence of nested Jordan curves whose sum of members of $z_n$ in the regions converges?

Suppose $(z_n)_{n\in\mathbb{N}}\subset \mathbb{C}$ such that $\displaystyle\sum_{n=1}^{\infty} z_n$ converges. Then, for example, $$ \sum_{k=1}^{\infty}\left( \sum_{\frac{1}{k+1} < \left\lvert z_n \...
Adam Rubinson's user avatar
3 votes
0 answers
67 views

An example of non simple group which is also a Lie group such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup

I want to know if there exist a non simple group (as abstractl group) $G$ such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup. I have tried some obvious examples like ...
Eloon_Mask_P's user avatar
0 votes
0 answers
48 views

On definition of Gâteaux derivative

Definition. Let $X,Y$ be Banach spaces and $f:U\rightarrow Y$ a map on an open subset $U\subset X$. The map $f$ is called Gâteaux differential at $x\in U$ if there exists a continous linear map $A:X\...
Hải Nguyễn Hoàng's user avatar
2 votes
0 answers
70 views

Are there contiguous sequences of prime numbers of length $k$ which are convex (similarly, concave) for every $k\in\mathbb{N}?$

Does the sequence of prime numbers contain contiguous subsequences of length $k$ which are strictly convex (similarly, strictly concave), for every $k\in\mathbb{N}?$ For example, $$ 17, 19, 23, 29 $$ ...
Adam Rubinson's user avatar
6 votes
2 answers
499 views

Looking for a counterexample for this statement regarding divergent series

I need counter for this statement : If the series $ \sum a_n$ is divergent then the series $b_n$= $\sum \text{min}(a_n,\frac{1}{n})$ is also divergent. I closest I reached to a counter is , Define $...
Gajjze's user avatar
  • 376
4 votes
1 answer
73 views

Does $\exists\ n$ such that the first $2n$ digits of Thue Morse, $X_{2n},$ is the concatenated sequence $X_n X_n?$ If not then why not?

Background: The Thue–Morse sequence is the binary sequence (an infinite sequence of $0$s and $1$s) obtained by starting with $0$ and successively appending the Boolean complement of the sequence ...
Adam Rubinson's user avatar
1 vote
1 answer
80 views

Is there a function $f : \mathbb{R}-\{ 0\} \to \mathbb{R}$ such that $f(x)= - f(2x)$ for all $x \in \mathbb{R}-\{ 0\}$ , $f(x) \ne 0$?

In my problem book there is this question Show by example that the condition $\lim\limits_{x \to 0} (f(x)+ f(2x) ) $ doesn't imply that $f$ has a limit at $0$. Although I solved this question but I ...
pie's user avatar
  • 3,591
0 votes
1 answer
68 views

How to find solutions of PDEs or explain they don't exist

Consider the equation $u_x + 3x^2y^2u_y = 0$ : Find a solution satisfying $u(x, 0) = 1/(1+x)$, or explain why no such solution exists. My attempt: I've managed to solve the equation and find the ...
starry41's user avatar
2 votes
1 answer
47 views

What are the compactness properties of $\mathbb R$, extended by a point with co-countable open neighborhoods?

In a remark in this answer regarding radiality and pseudoradiality in locally countable spaces, an example of a pseudoradial, countably tight space which failed to be sequential was constructed. The ...
M W's user avatar
  • 9,791
2 votes
1 answer
103 views

Example of two group filtrations that induce different topologies but the same completion

This is a question about completion in the sense of commutative algebra, i.e. inverse limit of quotients. Let $G$ be a topological abelian group, $G_0 \supseteq G_1 \supseteq \cdots$ and $H_0 \...
Anakhand's user avatar
  • 2,532
2 votes
1 answer
74 views

$m_i, n_j$ integers and $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'}.$ Does $\sum\frac{1}{m_i}=\sum\frac{1}{n_j}\implies\sum m_i\neq\sum n_j?$

Suppose $\{m_i\}_{i=1}^{k}$ and $\{n_j\}_{j=1}^{k'}$ are each finite subsets of $\mathbb{N},$ $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'},$ and $\displaystyle\sum_{i=1}^{i=k}\frac{1}{m_i} = \sum_{j=1}^{...
Adam Rubinson's user avatar
3 votes
0 answers
50 views

Counterexamples to $\hat{A} \otimes_A M \cong \hat{M}$

I'm looking for counterexamples to Proposition 10.13 in Atiyah-Macdonald when the hypotheses are not satisfied (throughout, $A$ is a ring, and $\hat{\phantom{M}}$ denotes the $I$-adic completion for ...
Anakhand's user avatar
  • 2,532
2 votes
0 answers
43 views

"Explicit" showing of non-flatness for a fractional ideal

I am not very familiar with flatness so I was trying to get a feel of what precisely is the crux of the notion, using objects I am used to. Sadly, most of the counter examples I could find where for ...
KeiOh's user avatar
  • 344
0 votes
0 answers
32 views

Counter example: continuity assumption of mean value theorem [duplicate]

Let's consider $f:[a,b]\to\mathbb{R}$ which is differentiable in each $x\in ~\!]a,b[$. Consequently $f$ is also continuous in those points. But we don't require that $f$ is continuous in $a$ and $b$. (...
Philipp's user avatar
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