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 ...

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Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
0
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1answer
15 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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2answers
37 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
1
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1answer
16 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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3answers
89 views

Examples of loops which have two-sided inverses.

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are ...
2
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2answers
52 views

Example of a semigroup with unique idempotent which is not a monoid

I am searching for an example of a semigroup, with unique idempotent element, that is not a monoid. Please help.
4
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2answers
782 views

Basic examples of monoids?

What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they ...
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10answers
6k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
2
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2answers
49 views

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$ The questions: Give an example of a nonempty subset of $\mathbb{R^2}$ (noted $M$) which is closed under addition and for all $m\in M$ we have $-m\in ...
0
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0answers
30 views

A question about compactly generated topology

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
288
votes
32answers
18k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
172
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25answers
16k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
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2answers
25 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 ...
34
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13answers
3k 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 ...
3
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2answers
174 views

Is there a bounded function discontinuous on a countable dense subset?

Is there a function that is continuous everywhere except on a countable dense subset, but is bounded? Bounded in the sense that supremum of $f$ is a finite number.
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8answers
412 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
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6answers
134 views

What would be an example of a magma such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?

Let $(M,\cdot)$ be a non-associative magma. What would be an example of $M$ such that there exists $x\in M$ such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?
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0answers
11 views

Vanishing cech cohomology of a concratible space.

Let $C$ be a concratible space. Is it possible to find a sheaf $F$ such that two Cech cohomology groups don't vanish? E.g. Can I find a sheaf (or a local system) and two integers $i,j$ such that ...
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1answer
20 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
3
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2answers
244 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 $2$?
2
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3answers
173 views

Basically disconnected space which is not extremally disconnected

Space $X$ is basically disconnected if every cozero-set has an open closure. Every extremally disconnected space is basically disconnected But i think the converse fails. The one-point ...
33
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18answers
6k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
13
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1answer
650 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 $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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1answer
36 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.
2
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2answers
82 views

An example of quasigroups with no identity

If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general). What would be some examples of a quasigroup without an identity ...
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2answers
74 views

Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈K$ exist?

Does there exist an algebraic structure $(K,+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in K$?
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1answer
23 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 ...
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2answers
31 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 ...
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0answers
31 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 ...
28
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0answers
479 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
3
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0answers
65 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 ...
3
votes
3answers
130 views

Need example for a topological space that isn't connected, but is compact

We know the topological space $(R,τ_1)$ is a connected space but it is not compact, $(R,τ_+)$ (which generated by $[a,b[$) is not connected space and it is not compact space, and $(R,τ_{cf})$ is ...
3
votes
4answers
80 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 $[0,1]$ ?
2
votes
1answer
57 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
34 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) ...
63
votes
9answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
7
votes
0answers
58 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 ...
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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 ...
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1answer
34 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
1
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3answers
66 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 $x_0$, ...
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 ...
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1answer
32 views

Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
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 ...
8
votes
1answer
326 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
6
votes
1answer
79 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 ...
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2answers
68 views

Compactness: Opens vs. Neighborhoods

Disclaimer: This thread is a record of thoughts. Discussion Given a compact set. Do mere neighborhood covers admit finite subcovers? $$C\subseteq\bigcup_{i\in I}N_i\implies C\subseteq ...
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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
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.
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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. ...
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1answer
66 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?