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)

3
votes
2answers
225 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
13
votes
1answer
622 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
10
votes
11answers
995 views

An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
1
vote
1answer
21 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
1
vote
2answers
30 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
0
votes
0answers
23 views

Nyquist–Shannon Sampling Theorem Counter Example?

I was learning about the Nyquist theorem regards signal processing the area of interest which I will rephrase below: Given a signal lasting infinitely long with a maximum frequency of f, then you can ...
0
votes
1answer
19 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
3
votes
4answers
76 views

A function not differentiable exactly two points of $[0,1]$. construction of such a function is possible?

Can a continuous function on $[0,1]$ be constructed which is not differentiable exactly at two points on $[0,1]$ ?
7
votes
0answers
53 views

Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
2
votes
1answer
51 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
32 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
52 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
158 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 ...
0
votes
3answers
64 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
25 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
26 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
156 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
78 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
42 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
28 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
26 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
32 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 $f'(x)=|x|$ ...
4
votes
8answers
400 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
99 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
27 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
33 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
40 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
22 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
36 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
61 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
72 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
105 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
95 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 ...
1
vote
4answers
73 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
20 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 his torsion. I ...
4
votes
0answers
42 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
33 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
156 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
50 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
44 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
56 views
+100

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
44 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
95 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: ...
5
votes
1answer
79 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
44 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
37 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
47 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
77 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
106 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 ...
-1
votes
1answer
34 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 ...