Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

learn more… | top users | synonyms (2)

2
votes
1answer
72 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
2
votes
2answers
59 views

counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
2
votes
1answer
66 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
3
votes
2answers
162 views

Is the converse of Sylvester's inertia law true?

Let $A,B$ be $n\times n$ symmetric matrices. Assume the positive,negative,0 index of $A$ and $B$ are the same. (That is, they have the same inertia) Then, are $A,B$ congruent? What is a ...
3
votes
3answers
72 views

Example for why L'Hospital's rule demands that the denominator is locally not $0$.

We're looking at the limit of a function $f(x):=\frac{g(x)}{h(x)}$ for $x \rightarrow x_0$. Since L'hospital's Rule demands that the denominator function $h(x) \neq 0$ for a neighbourhood of $x_0$, ...
1
vote
2answers
154 views

Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to ...
0
votes
1answer
29 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
1
vote
2answers
161 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...
6
votes
1answer
92 views

An example of a simple infinite $2$-group

Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian. Take the subgroup ...
0
votes
2answers
43 views

Linear algebra questions

$M$ and $N$ subspaces of $V$. Give examples of $M$ and $N$ such that $M\cup N$ and $M\setminus N$ are not subspaces.
0
votes
1answer
29 views

Proof by counter-example

As is probably obvious by the title I cannot work out any set of numbers which disporoves the following conjecture $$ a^2 > b^2 => a > b $$ where A and B are real numbers. Anyone to give a ...
2
votes
0answers
39 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
1
vote
1answer
52 views

Differentiable continuous function whose derivative is not continuous [duplicate]

Is there a function which is continuous and differentiable, but is not smooth function? By smooth I mean having continuous derivative. For example, the derivative of $f(x)=x|x|/2$ is $f'(x)=|x|$ ...
5
votes
8answers
455 views

Does a non-abelian semigroup without identity exist?

I was introduced to semigroups today and had a question. So all the examples of semigroups I was given were either monoids or groups. So I was curious, does there exist a semi-group which is not ...
2
votes
1answer
106 views

Weaker definitions of Lie subgroups

A Lie group $H$ is called a Lie subgroup of a Lie Group $G$ if there is a map $i:H\to G$ which is (a) an injective immersion and (b) a group homomorphism. My questios are: What happens if we replace ...
0
votes
1answer
38 views

Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties: Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of ...
2
votes
2answers
37 views

Groupoid with division

I'm searching an example of a grupoid with division which is not a quasigroup. A grupoid $(G, \cdot)$ is with division if $a\cdot G=G\cdot a=G$. I was thinking to try $(\mathbb{Q},\cdot)$, where ...
0
votes
1answer
43 views

How to find example such $A^{(n)}$ are different each other

let denoted $A'$ Derived set,and define $$A'=A^{(1)},(A^{(1)})'=A^{(2)},\cdots,(A^{(n)})'=A^{(n+1)}$$ Question: Take example the set $A$,such $A^{(k)},k=1,2,\cdots,n+1$ are different ...
1
vote
1answer
35 views

Example of a unbounded projection

Let $H$ be a Hilbert space over $\mathbb{K}$. Let $T:H\rightarrow H$ be a linear transformation such that $T^2=T$. What is an example of $T$ such that $T$ is unbounded?
1
vote
2answers
38 views

A question about the orders of the elements of a group [duplicate]

Let $m$ and $n$ be to positive integers strictly larger than $1$. Is it possible to find a group $G$ in which there are two elements, say $a$ and $b$, such that the order of $a$ is $m$, the order of ...
2
votes
0answers
100 views

An example of wreath product

I was analyzing the following example of wreath product of groups. Let $\mathbb{Z}_2$ be the cyclic group of order two and $\mathbb{Z}$ be the usual additive group of integers. Consider the ...
1
vote
2answers
77 views

Exactness of $dx,dy$

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is: Does there exist a space such that either $dx$ or $dy$ or both are exact? (A ...
6
votes
1answer
119 views

Prove or disprove that ${F_{n}}^2 + 41$ is always a composite

The problem: Prove or disprove: If $F_{n}$ is the $n^{th}$ Fibonacci number then $${F_{n}}^2 + 41$$ is always a composite number. It looks that if $n$ is not multiple of 12, ${F_{n}}^2 + 41$ ...
6
votes
1answer
104 views

Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
2
votes
4answers
75 views

Example where $\lim\limits_{m\rightarrow\infty} \int_E f_m =0$ and $f_m(x) \not\rightarrow 0$

I am looking for an example of a sequences of non-negative and measurables functions with $\lim\limits_{m\rightarrow\infty} \int_E f_m =0$ and $f_m \nrightarrow 0 \:\:\forall\:x\in E$
1
vote
2answers
27 views

Non exact sequence of quotients by torsion subgroups

$$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion ...
4
votes
0answers
45 views

about the closure of by pointwise convergence of sequences.

In the book "Integration Theory" (LNM315, K.Bichteler, p.65) a family $\mathcal{F}$ of real function is called full if it is close by pointwise convergence of dominated (by some element of ...
1
vote
2answers
52 views

Does every inner product space have an orthonormal basis?

I'm reading Halmos' text and he defines 'basis' as a maximal orthonormal subset of a Hilbert space $H$, but this definition seems inconsistent with the standard definition of basis. With the standard ...
13
votes
4answers
199 views

A continuous function on $[0,1]$ not of bounded variation

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such ...
2
votes
4answers
66 views

Examples of magmas with all their elements idempotents

A magma is supposed to be closed under a binary operation. Are there examples of magmas with all their elements idempotents under the operation of the magma?
0
votes
1answer
60 views

Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
3
votes
0answers
66 views

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
1
vote
1answer
47 views

Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} ...
5
votes
1answer
97 views

Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
6
votes
1answer
90 views

Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent?

Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$. Is the following series always ...
0
votes
1answer
46 views

What is an example such that $f(x)\neq \sum_{m=0}^{k-1} \frac{f^{(m)}(\alpha)}{m!} (x-\alpha)^m + \frac{f^{(k)}(\psi)}{k!} (x-\alpha)^k$?

Let $f:[a,b]\rightarrow \mathbb{C}$ be a $C^{k-1}$ and assume $f^{(k-1)}$ is differentiable on $(a,b)$. If the range of $f$ is real, then the usual taylor's theorem holds, but I'm not sure whether ...
1
vote
1answer
42 views

Topological Spaces: Pre-Uniform Structures

Disclaimer This thread is meant to record. See: Answer own Question Reference It is a follow-up to: Uniform Spaces: Neighborhood System It has relevance to: TVS: Uniform Structure Problem Given ...
2
votes
0answers
69 views

Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
2
votes
4answers
89 views

Topological Spaces Involving Connectedness, Compactness, and Hausdorfness

I made a table about topological spaces with or without connected, compact, and Hausdorff properties. However, I cannot find the example for the following cases: Compact, but neither Hausdorff nor ...
4
votes
0answers
125 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
0
votes
1answer
94 views

Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
2
votes
1answer
50 views

Is every open set in a base space evenly covered?

Let $C$ be a covering space of $B$. Then, does every open set in $B$ evenly covered by a covering map? This must be false but I cannot find a counterexample.. Please help
2
votes
0answers
69 views

Counterexample to an implication

Denote $\bar{A}$ a complement of $A$ in a set $\Omega$ and $A \Delta B = A/B \cup B/A$ the symmetric difference of $A, B$. It is claimed that for a map $\phi := \Omega \rightarrow \lbrace 0, 1 ...
2
votes
1answer
51 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
1
vote
3answers
56 views

Find two functions $f$ and $g$ such that they are both discontinuous at $c$, however, $f+g$ and $f\cdot g$ are both continuous at $c$

Could someone please explain to me how to approach these kinds of question and also what is the answer to the following question? Give an example of a function $f$ and $g$ such that they are both ...
1
vote
0answers
34 views

Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
0
votes
2answers
42 views

The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...
4
votes
2answers
90 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
1
vote
1answer
59 views

Show that the space of superharmonic functions is not a linear space

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$. A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ ...
1
vote
1answer
52 views

Open (but not closed) subgroups of $GL_n$

The book I am currently reading states: "...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds." Prompted me to think about open subgroups of $GL_n$: ...