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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Are there examples where torsion subgroup of the first homology group is not $(\mathbb{Z}_2)^n$?

I've been trying to understand nontrivial torsion subgroups $G$ of the first homology group $H_1\cong\mathbb{Z}^s \times G$, where $s$ is the rank (first betti number) and G is a finite abelian group. ...
Diana's user avatar
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Examples of "smooth" sets in $\mathbb{R}^2$, $\mathbb{R}^3$

What is the most simplest straight forward example of a set in $\mathbb{R}^2$ or $\mathbb{R}^3$ which has "smooth" boundary? Where $H(Q_+)=U\cap Q$ needs to be replaced by $H(Q_+)=U\cap \...
Perelman's user avatar
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2 votes
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68 views

Behaviour of $a^k\pmod b,\ k=1,2,3,\ldots$

Let $n\in\mathbb{N}.$ Suppose $\left(x_k\right)_{k=1}^{n}$ is a $k-$tuple of $-1$'s and $1$'s. Let (the function) $r(i,j)$ be the remainder when $i$ is divided by $j,$ so that $0\leq r(i,j) \leq j-1.$ ...
Adam Rubinson's user avatar
3 votes
2 answers
107 views

Why does this proof work: $\sum\limits_{n=1}^ \infty \left(\frac{1}{4n-1} - \frac{1}{4n}\right)= \frac{\ln(64)- \pi}{8}$?

$$f(x):= \sum_{n=1}^ \infty \left(\frac{x^{4n-1}}{4n-1} - \frac{x^{4n}}{4n}\right)$$ $$f'(x) = \sum_{n=1}^ \infty ( x^{4n-2}- x^{4n-1})= \frac{x^2}{(1+x)(1+x^2)}$$ $$\int_0 ^1 \frac{x^2}{(1+x)(1+x^2)}=...
pie's user avatar
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1 answer
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Does there exist a group $G$ such that $(G\setminus S)^2=G$ for each $S\subseteq G$ with $|S|<\infty$?

Note: I have answered this question myself in typing it up. I thought I'd share it because it took me too long. Maybe someone will benefit from it. The Question: Does there exist a group $G$ such ...
Shaun's user avatar
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Consequence (if true) of $\gcd(n!+1,n^2+1)=1$ for $n>1$ an integer . The Buniakowsky's conjecture.

It's a follow up of Do we have $\sum_{n=1}^{\infty}\frac{\gcd\left(1+n!,1+n^{2}\right)}{n!}\stackrel{?}{=}e$? Context/ I want to find a consequence of this conjecture which could resists perhaps for ...
Miss and Mister cassoulet char's user avatar
1 vote
0 answers
71 views

Is there any other way to make a $f :[0,1]\to \mathbb{R}$ discontinuous at every point of its domain?

The Dirichlet function: $$f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational } \end{cases}$$ is an example of a function that is discontinuous everywhere ...
pie's user avatar
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Is There any Untraceable Generalized Petersen Graph?

The Petersen graph is one of the example of graph which is not Hamiltonian. Can we find an example among the generalized Petersen graph which doesn't have Hamiltonian path (untraceable)?
user966's user avatar
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Examples of sequent derivations that uses cut rule that can be modified to not to use cut rule?

The cut-elimination theorem states that any sequent calculus derivation that uses the cut rule also has a derivation that does not use the cut rule. I cannot find any explicit examples of such ...
John Davies's user avatar
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33 views

Find necessary and sufficient conditions for ordinal monotonicity.

First of all let's we remember the following result. Theorem Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ ...
Antonio Maria Di Mauro's user avatar
1 vote
3 answers
90 views

Find an equivalence relation over all of $\mathbb{Z}$ which has infinitely many equivalence classes with infinitely many elements in each

I want to find an equivalence relation defined on all integers (that is, all of $\mathbb{Z}$) where The equivalence relation partitions $\mathbb{Z}$ into infinitely many equivalence classes; and ...
Christopher Miller's user avatar
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1 answer
28 views

Is there a differentiable $f$ st $f' \ne 0$ st for $a\not \in \mathbb{Q}, \ f(a) \in \mathbb{Q}$?

There are many easy examples for differentiable $f$ st $f'(x) \ne 0$ and for $a \in \mathbb{Q}, \ f(a) \not \in \mathbb{Q}$ for example $\pi x , e^x,$ etc, but the question is: Is the converse true ?...
pie's user avatar
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-1 votes
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Almost sure convergence do not imply convergence in mean [closed]

What is the counterexample to show that almost sure convergence does not imply mean (L^p space) convergence? All the examples I've seen were about the fact that convergence in mean does not imply ...
myfakeaccount's user avatar
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1 answer
30 views

Seeking Counterexamples: Bilinear Maps Continuous in Components but Not Globally in Non-Complete Normed Spaces

It can be shown that when X, Y, and Z are all Banach spaces (or at least when X or Y are Banach spaces) over the number fields R or C, and when B : X×Y →Z is a bilinear function, the continuity of B ...
Matrix AC's user avatar
-1 votes
1 answer
74 views

What's wrong with applying our intuition for the behavior of objects in low dimension to high dimension [closed]

The following text is taken from the a book about linear programming that I'm reading: A graphical illustration is useful for understanding the notions and procedures of linear programming, but as a ...
Tran Khanh's user avatar
2 votes
2 answers
186 views

An easier example of a non-PID where every finitely generated ideal is principal

Say that an integral domain $\mathcal{X}$ is an almost-PID if $\mathcal{X}$ is not a PID but every finitely generated ideal of $\mathcal{X}$ is principal. The question of whether almost-PIDs exist ...
Noah Schweber's user avatar
4 votes
1 answer
47 views

$A\subset\mathbb{N},$ natural density $1/2.$ Half the members of $A$ are even, half are odd. Is $A$ an (eventual) additive basis of $\mathbb{N}?$

Proposition: If $A\subset\mathbb{N},\ A$ has natural density $d> \frac{1}{2},$ then $\exists\ N\in\mathbb{N}\ $ such that $\ n>N \implies \exists\ a,b\in A\ $ such that $\ a+b=n.$ Proof sketch: ...
Adam Rubinson's user avatar
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0 answers
35 views

Zeno's Monoid: Has anyone got a reference for this?

Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F: x*y = x+y - xy/k Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element. Moreover ...
Nicholas Bamber's user avatar
3 votes
1 answer
118 views

Given a tuple of $k$ distinct integers, is there a generator list in a $\mathbb{Z}/n\mathbb{Z}$ that matches the tuple?

Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1).$ Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all nonzero) does $\exists$ integers ...
Adam Rubinson's user avatar
0 votes
2 answers
42 views

Example of non isomorphic field extensions

What is an example of non isomorphic field extensions? I would like such a non example as I think it would explain the significance of the commutativity of the square condition in an isomorphism of ...
cpt.price's user avatar
1 vote
1 answer
69 views

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
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$\dim(U_1 \cap U_2 \cap U_3) = n − 3$, 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
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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
96 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
  • 9,685
4 votes
3 answers
112 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
401 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
107 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
180 views

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
78 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
  • 1,204
1 vote
0 answers
26 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
35 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
0 votes
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
  • 11
0 votes
2 answers
81 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
0 votes
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
103 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
195 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
62 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
  • 1,115
2 votes
1 answer
85 views

If $f,g$ are uniformly continuous and $f$ is bounded and non-periodic, then $fg$ is not necessarily uniformly continuos

I've just begun my grad program and we were introduced to this problem in our Analysis I course: consider two uniformly continuous functions $f$ and $g$, say from $\mathbb R$ to $\mathbb R$, where $f$ ...
Arthur's user avatar
  • 53
0 votes
0 answers
51 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
  • 1,626
1 vote
1 answer
89 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
  • 3,925
3 votes
1 answer
128 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
4 votes
2 answers
282 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
  • 7,048
1 vote
1 answer
193 views

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
144 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
35 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
  • 2,572
1 vote
0 answers
34 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 $...
User2427's user avatar
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