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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
37 views

If $f:X\longrightarrow Y$ is surjective, closed and continuous then $Y$ is weakly locally connected provided $X$ is it.

First of all let's we give the following definition. Definition A space $X$ is weakly locally connected if for each neighborhood $N_x$ of any $x\in X$ there exist a connected subspace $F_x$ and a ...
1 vote
1 answer
69 views

Can set of complex numbers be a subring of the ring of quaternions?

I was studying an algebra book by Dummit and Foote, there I saw an example of subrings; $\mathbb H_{\mathbb Z}=$ $\{a+bi+cj+dk: a,b,c,d\in\mathbb Z\}$ is a subring of $\mathbb H_{\mathbb R}$=$\{a+bi+...
2 votes
1 answer
38 views

Are there any graph theory results which clearly fail for non-simple graphs?

Many times when I do graph theory, I might write something like "assume for simplicity that $G$ is a simple graph", thinking that this doesn't really pose a major problem and some elementary ...
1 vote
1 answer
25 views

Give an example of a ring $A$ and a multiplicative set $S$of $A$ such that $S^{-1}A$ is a PID (but not a field), but $A$ is not a PID.

I know that if $A$ is a PID and $S\subset A^*$ is a multiplicative set. Then $S^{-1}A$ is a PID since ideals of $S^{-1}A$ is of the form $S^{-1}I$ where $I$ is an ideal of $A$. We have to find a ...
  • 2,663
1 vote
1 answer
73 views

if $V\subseteq U\subseteq V\cup W$ then is it true that $U\subseteq W$?

So I am trying to understand if $U$, $V$ and $W$ are three sets such that $$ \tag{0}\label{0}V\subseteq U\subseteq V\cup W $$ then the inclusion $$ \tag{1}\label{1}U\subseteq W $$ must hold: indeed, ...
0 votes
1 answer
17 views

Equicontinuity of a set of functions: contrexemple

Let $E$ be a Banach space, and $I=[0,1]\subseteq \mathbb{R}$. My goal is to construct a set of functions $X\subseteq\mathcal{C}(I,E)$ such that: $X$ is equicontinuous but not relatively weakly compact....
  • 702
0 votes
0 answers
20 views

If $d(p_n,q_n)$ converges to $0$ then a limit point for the the range of $p_n$ is a limit point for the range of $q_n$, provided $p_n$ is injective.

Let be $p$ and $q$ are two sequences in a metric space $(X,d)$ and thus let's we prove that if the sequence $d(p_n,q_n)$ converges to zero then the limite point of $$ P:=\{p_n:n\in\omega\} $$ are ...
17 votes
1 answer
247 views
+500

If two graphs have same number of trees of every kind, must they be isomorphic?

Set-up. Let $G$ be a (simple) graph. Given a tree $T$, let us define: $$ a_{T}(G) = \text{number of subgraphs of } G \text{ that are isomorphic to } T $$ If $T$ and $T'$ are isomorphic, then of course ...
  • 10.1k
1 vote
1 answer
46 views

Are there sequences in $\mathbb{R}^2$ which converge monotonically in a strong sense?

Let $x_n \in \mathbb{R}^n$ be a convergent sequence whose limit is $x$. (Invented) definition: We say that $x_n$ converges monotonically to $x$, if for every open connected neighbourhood $U$ of $x$, ...
0 votes
0 answers
16 views

Is the Jordan inner measure different from the Lebesgue inner measure?

As Tao explains in his blog: "one does not gain any increase in power in the Jordan inner measure by replacing finite unions of boxes with countable ones. But one can get a sort of Lebesgue inner ...
  • 4,177
4 votes
1 answer
96 views

Is it possible for a convex function that is not strictly convex to be nowhere linear?

We say that a function $f\colon (a,b)\to \mathbb{R}$, where $-\infty\leq a<b\leq +\infty$, is convex (resp. strictly convex) if for all $x,y\in (a,b)$ with $x<y$ and for every $t\in (0,1)$ it ...
  • 235
2 votes
0 answers
40 views

A counterexample about orthogonal complement.

In our functional analysis course we have shown that if $H$ is a Hilbert space and $M$ is a closed subspace then $H=M\oplus M^{\perp}$.However I know that this is not necessarily true if $M$ is not ...
4 votes
2 answers
100 views

If $(a_n)_n,\ (b_n)_n,$ are positive convex decreasing sequences, $\sum a_n$ converges and $\sum b_n$ diverges, then $\ \frac{a_n}{b_n}\to 0.$

Defintition: A real sequence $\ (x_n)_n\ $ is convex if $\ x_n - x_{n+1} \geq x_{n+1} - x_{n+2}\quad \forall\ n\in\mathbb{N}. $ Continuing on from this question here, Proposition $\ 3:\ $ If $\ (a_n)...
2 votes
1 answer
60 views

If $(a_n),\ (b_n)$ are positive decreasing sequences, $(a_n)$ is convex, $\sum a_n$ converges and $\sum b_n$ diverges, then $\frac{a_n}{b_n}\to 0.$

Proposition $\ 1:\ $ If $\ (a_n)_n,\ (b_n)_n,\ $ are positive decreasing sequences such that $\ \displaystyle\sum a_n \ $ converges and $\ \displaystyle\sum b_n \ $ diverges, then $\ \frac{a_n}{b_n}\...
0 votes
0 answers
52 views

example for $K\rtimes H\cong G\rtimes H$ and $K\ncong G$

Let $H,L$ be groups acting on $K,G$. Show that $K\rtimes H\cong G\rtimes L$ and $K\ncong G$ is not true in general. I found a solution online using $\times$: Counterexample: $G \times K \cong H \times ...
2 votes
1 answer
55 views

Outer automorphisms of a holomorph

Let $G$ be a finite group. Although $G$ may have outer automorphisms, an easy way to "kill" them is taking the holomorph, $G\rtimes\mathrm{Aut}(G)$. However, this process may a priori ...
  • 11.5k
3 votes
0 answers
60 views

If $(a_n)$ is decreasing and $\sum a_n$ diverges. Then for each increasing subsequence $(n_k)\subset\mathbb{N},\sum n_k(a_{n_k}-a_{n_{k+1}})$ diverges

I had a proposition: Suppose $\ (a_n)_n\ $ is a positive, decreasing real sequence such that $\ \displaystyle \sum_n a_n\ $ diverges. Then for any increasing subsequence $\ (n_k)_k\ \subset \mathbb{N}...
1 vote
0 answers
40 views

Inclusion of classical spaces in analysis

Context: Today I was introduced to a very nice diagram where we can see the relation between several spaces in analysis. It goes as follows: $$\text{Hilbert Sp.} \subset \text{Inner Prod. Sp.} \subset ...
  • 131
0 votes
1 answer
27 views

A normed space in which unit circle in convex

Let $X$ be a normed linear space (real or complex). Consider $B = \{x \in X : \lVert x \rVert = 1\}.$ I want an example of a non-trivial normed space in which $B$ is a convex set. I have proved that $...
  • 78
-2 votes
1 answer
31 views

Which is an example of a real function satisfying conditions 1. and 2.? EDITED

I am trying to find an example of function $f:\mathbb{R}\to\mathbb{R}$ satisfying: there exists $c>0$ such that $$\lim_{|x|\to\infty} |f(x)|<c$$ and there exist $\varepsilon, c_1>0$ and $\...
  • 2,735
2 votes
0 answers
35 views

Function defined differently on rationals and irrationals with derivative given by classic rules

Today, one of my students was given the function $$f(x) = \begin{cases} x^3+1 \quad \text{ if }x\in \Bbb Q\\ x^3+x \quad \text{ if }x\in \Bbb R \backslash \Bbb Q \end{cases} $$ He had some questions ...
  • 1,210
0 votes
0 answers
12 views

Why do we need the AEP algorithm? The discrete case

I recently found a paper describing the AEP algorithm, an algorithm to compute $$P(X_1 + X_2 + \dots + X_d \leq s)$$ , given the joint distribution of $(X_1, \dots, X_d)$. O.k., why do we need this ...
1 vote
0 answers
49 views

Prove or disprove that if $f:X\longrightarrow Y$ is a perfect map then $w(Y)\le w(X)$

Let be $f$ a perfect map from a space $X$ to a space $Y$ so that I am trying to prove or disprove if the inequality $$ \begin{equation}\tag{1}\label{1}w(Y)\le w(X)\end{equation} $$ holds, where $w$ is ...
0 votes
0 answers
31 views

( countability axiom )Let $\tau$ and $\sigma$ be topologies over a set $X$ such that $\tau \subseteq \sigma$...

Let $\tau$ and $\sigma$ be topologies over a set $X$ such that $\tau \subseteq \sigma$. a) If $(X, \tau )$ satisfies a certain countability axiom then $(X, \sigma)$ also satisfy? b) If $(X, \sigma)$ ...
  • 127
-1 votes
2 answers
84 views

Give a group $G$ such that ${\rm Aut}(G)/{\rm Inn}(G) \cong \mathbb{Z}_2$. [closed]

Give a group $G$ such that ${\rm Aut}(G)/{\rm Inn}(G) \cong \mathbb{Z}_2$. So far the only example I've seen of this is with $G = S_6$ but honestly I just don't get the proof for this case so I was ...
  • 141
4 votes
1 answer
63 views

Counter-example: approximate an integrable function from below by a continuous function with compact support

I'm reading this thread in which the OP asked if the following statement is true, i.e., Let $f$ be a non-negative measurable function and suppose that $f\in L^p(\Bbb R^n)$, then for any $\delta \gt 0$...
  • 2,943
1 vote
0 answers
18 views

Example of a (non-smooth) algebraic group $G$, a normal subgroup $N$ and a representation such that the sum of $N$-eigenspaces is not $G$-stable

Let $k$ be an algebraically closed field and define algebraic groups to be affine $k$-group schemes of finite type. Does anyone know of an example of an algebraic group $G$, a normal algebraic ...
3 votes
1 answer
29 views

If $f$ is a function from $X$ to $Y$ then does the inclusion $f^{-1}\big[Y\setminus f[X\setminus U]\big]\subseteq U$ hold for any $U\in\mathcal P(X)$?

Given a function $f$ from $X$ to $Y$ I am trying to prove or to disprove the inclusion $$ \tag{1}\label{1}f^{-1}\big[Y\setminus f[X\setminus U]\big]\subseteq U $$ where $U\in\mathcal P(X)$. So I ...
2 votes
1 answer
29 views

If $f$ is a function from $X$ to $Y$ then does the equality $f[X]\setminus f[X\setminus U]=f[U]$ holds for any $U\in\mathcal P(X)$?

Given a function $f$ from $X$ to $Y$ I am try to prove or to disprove the equality $$ \tag{1}\label{1}f[X]\setminus f[X\setminus U]=f[U] $$ where $U\in\mathcal P(X)$. So I surely know thath for any $U,...
0 votes
0 answers
30 views

Discussion for understanding existence and uniqueness theorem.

We have been taught the following version of existence and uniqueness theorem in ordinary differential equations: Theorem: Let $D\subset \mathbb R^2$ be an open connected set and let $f:D\to \mathbb ...
2 votes
2 answers
175 views

There exists an function $f$ with this properties?

Let $N >2$ be an fixed natural number. There exist an function $f$ such that? $(1)$ $f \in C^{1}(\mathbb{R};\mathbb{R}),$ $(2)$ $\lim_{s \to 0} \frac{f(s)}{s}=0$ $(3)$ There exists $C>0$ and $p \...
  • 192
0 votes
1 answer
27 views

Help with counter-example using conditional expectations

I am trying to construct a counter-example for a certain problem in a Statistics class. Right now, to finish the counter-example, I just need to find three random variables, let's say, $(X, Y, Z)$, ...
0 votes
0 answers
17 views

Is $p^*$ the best integrability we can get from $W^{1,p}$?

The Gagliardo Nirenberg Sobolev (GNS) inequality tells us that $$ \exists c > 0,\quad \forall u \in W^{1,p}(\mathbb R^d),\quad ||u||_{L^{p^*}} \leq c ||\nabla u||_{L^p} $$ whenever $1 \leq p < d$...
  • 1,544
2 votes
1 answer
35 views

Holomorphic functions with finite fibers

Suppose $f\colon\mathbb C\to\mathbb C$ is holomorphic and injective. Then, $f$ is a polynomial (and in fact, linear,) by observing that otherwise $f$ has an essential singularity at $\infty$ so $f(\{z\...
  • 11.5k
1 vote
1 answer
72 views

If $d_1$ and $d_ 2$ are two distances in X then are the respective topologies equivalent when $d_ 1$ and $d_2$ are continuous with respect each other?

Let be $d_1$ and $d_2$ two distances in a set $X$ and thus let's we suppose that they are continuous with respect each other, that is $d_1$ is continuous with respect the product topology generated by ...
1 vote
2 answers
49 views

Does we have $\int_{\sum_{i=1}^{\frac{p}{k}}\frac{1}{i}}^{\sum_{i=1}^{p}\frac{1}{i}}e^{-\left(\frac{x}{p}\right)^{p}}dx\to \ln(k)$ as $p\to \infty$?

Problem/Conjecture : Let $p,k,n\in(3,\infty)$ be positive integers such that $p=kn$ then it seems we have as $p\to \infty$ : $$\int_{\sum_{i=1}^{\frac{p}{k}}\frac{1}{i}}^{\sum_{i=1}^{p}\frac{1}{i}}e^...
  • 3,471
1 vote
0 answers
68 views

Total variation distance for product measures

Consider two measurable spaces $X$ and $Y$ and form their Cartesian product $X\times Y$, with the product $\sigma$-algebra. In this question I asked whether, given probability measures $p$ and $q$ on ...
  • 7,395
0 votes
0 answers
53 views

Examples-Counterexamples for understanding Fubini's theorem.

Fubini's theorem in measure theory states that, Statement: Let $(X,\mathcal S,\mu)$ and $(Y,\mathcal F,\nu)$ be two $\sigma$-finite measure spaces,and $\mu\times \nu$ denote the product measure on $X\...
1 vote
1 answer
17 views

Minimum but not positive definite neighborhood

Let $f \in C^2(\mathbb{R})$. It is clearly the case that $\bar x \in \mathbb{R}^n$ is a local minimizer if $\nabla f(\bar x)=0$ and there exists some $\varepsilon > 0$ such that $$ H f(x) \text{ is ...
  • 5,324
2 votes
0 answers
35 views

Disintegration theorem: counter-examples in which $\mu_y$ is supported on $\pi^{-1} (y)$ not for $\nu$-a.e. $y\in Y$

Recently, I came across Tao's blog about disintegration theorem. Disintegration theorem Let $X$ be a compact metric space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability ...
  • 2,943
1 vote
1 answer
24 views

Example of an endomorphism on space of continuous functions with infinite norm

Thinking about endomorphisms with infinite norms on infinite dimensional spaces, I tried to come with an example on $C(\mathbb {R} )$, and I am confused: would endomorphism $\phi: f(x) \rightarrow x^2 ...
0 votes
0 answers
123 views

Very high unique counterexamples in mathematics [duplicate]

There are conjectures that have very high counterexamples, but a plethora of them. What are some not-overly-specific conjectures about a natural number $n$ that have only one high counterexample. (I ...
  • 2,598
1 vote
0 answers
26 views

If P, Q are invertible matrix, show the following Moore Penrose inequality $(PAQ)^+ \neq Q^{-1}A^+P^{-1}$ with a counterexample.

If P, Q are invertible matrix, show the following Moore Penrose inequality $(PAQ)^+ \neq Q^{-1}A^+P^{-1}$ with a counterexample. $A^+$ is the Moore Penrose Inverse. I have tried many examples but I ...
3 votes
1 answer
100 views

Can every manifold be embedded into a compact manifold of the same dimension

Can every connected smooth boundary-less manifold be embedded into a compact smooth boundaryless manifold of the same dimension ? If not, can someone please provide me with a counterexample ? Thank ...
  • 19.6k
3 votes
1 answer
57 views

Is there a normed space $W$ such that $W=X\oplus Y$, with $X$ not closed and such that $W$ is homeomorphic to $X\times Y$?

If $W$ is a normed space, $X,Y$ are vector subspaces with the subspace topology, $X$ is not closed and $X\oplus Y=W$ in the algebraic sense, then the map $X\times Y\to W$, $(x,y)\mapsto x+y$ is ...
0 votes
1 answer
123 views

Folland Chapter 7, Exercise 13

Chapter 7 Exercise 13 of Folland's "Real Analysis" reads Let $X=\mathbb R\times \mathbb R_d$ where $\mathbb R_d$ denotes the discrete topology. Let $f$ be a function on $X$ and define a ...
  • 6,527
5 votes
0 answers
72 views
+50

Is every function eligible as a boundary condition?

Consider the Laplace equation on unit square domain: $$ u_{xx} + u_{yy} = 0, 0 < x < 1, 0 < y < 1 $$ I wondered whether there is any boundary condition that wouldn't let this equation have ...
  • 1,337
7 votes
3 answers
154 views

Mutually independent vectors with small, integer coefficients

Consider a family $\cal V$ of $n$ vectors $v_1,\ldots,v_n$ in ${\mathbb N}^3$ (here ${\mathbb N}$ denotes the set of positive integers, excluding $0$). Say that $\cal V$ is strongly independent if any ...
  • 59.9k
2 votes
1 answer
45 views

Convex optimization problem not expressible as a conic program

I've been reading Boyd & Vandenberghe and it says that conic programming is a subclass of convex optimization. I haven't been able to find an example of a convex optimization problem that I cannot ...
  • 457
0 votes
0 answers
30 views

Finite order of a distribution

[Exercise about distribution theory; definition of a distribution and examples:] Let $\Omega \subset \mathbb R^{n}$,$f \in L^{1}_{loc}(\Omega)$, $ \beta \in \mathbb{N}^{n}$ multi-index. Consider the ...

1
2 3 4 5
102