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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3answers
150 views

Examples of quasigroups with no identity elements

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 quasigroups without an identity ...
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1answer
30 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
2
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0answers
38 views

Not continuous function with closed graph

I would like to see an explicit example of a function $f:R\rightarrow R$ which has a closed graph in $R^2$ but is discontinuous at every point in the real line.
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1answer
34 views

Closed Subsets of the Real Line that are Uncountable

If a subset of the real line is uncountable and closed, does it have to contain a closed interval? Is there any theorem related to this?
2
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1answer
34 views

Is this an example of a sequential non-Fréchet–Urysohn space?

Let $X$ be the set $X = \{ (0,0) \} \cup \{ (\frac{1}{n},0) : n \in \mathbb N \} \cup \{ (\frac{1}{i},\frac{1}{k}) : i,k \in \mathbb N \}$. Points of the form $(\frac{1}{i},\frac{1}{k})$ are ...
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1answer
103 views

Example of $\deg(fg)<\deg(f)+\deg(g)$

Let $R$ be an integral domain and $f,g\in R[X_1,...,X_n]$ where $n>1$. What is an example of a pair $f,g$ such that $\deg(fg)<\deg(f)+\deg(g)$? Moreover, i have proven that the units of ...
2
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0answers
48 views

specific magma examples

Give an example of a magma $S$ such that $S$ has a zero and $S$ has a left zero divisor that is not a right zero divisor an example of a magma with an identity such that there is an element with ...
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votes
1answer
24 views

Oscillating essential discontinuities exist?

Let $f$ be a function $\mathbb R \to \mathbb R$. According to Wikipedia an discontinuity of $f$ is essential if and only if either the left or the right limit is infinite or does not exist. Is it ...
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1answer
57 views

Continuity and interior

I have questions about the relation between continuity and interior based on the article ;Continuity and Closure At first I guess that there will be a property like $f:X\rightarrow Y$ is continuous ...
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3answers
1k 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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0answers
37 views

Example of a function such that iterated integrals are equal

Is there any example of a function $f(x,y):[0,1]$x$[0,1]\to \mathbb R$ so that $\int_{0}^1\int_{0}^1f(x,y)dydx$ and $\int_{0}^1\int_{0}^1f(x,y)dxdy$ exists and are equal but $\int\int f(x,y)dydx$ does ...
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1answer
107 views

Examples for 1d finite element method

I am looking for some examples for my Finite Elements project (in one dimension). I have written code in MATLAB and would like to show a few examples of it working. I have one or two general examples ...
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2answers
41 views

Non-Hausdorff space such that all connected components are singletons

Is there a topological space $(X,\tau)$ such that $(X,\tau)$ is not Hausdorff; if $S\subseteq X$ and $S$ contains more than 1 point, then $S$ is not connected (with the subspace topology).
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0answers
39 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
2
votes
1answer
74 views

Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
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votes
2answers
110 views

Product of quotient maps and quotient space that is not Hausdorff

I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff.
0
votes
1answer
77 views

Normal subgroup question/example [duplicate]

Give an example(s) of a group $G$ and $H\leq G$ (i.e., $H$ is a subgroup of $G$) where $H$ is not normal in $G$. What about $S_3$ and $\langle(1\;2)\rangle=\{(1),(1\;2)\}$? [Since ...
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2answers
139 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
1
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2answers
70 views

Identity about a Functor

I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind: Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for ...
2
votes
2answers
204 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
0
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1answer
37 views

Example of a subgroup that is not normal (not involving permutations)

It would be great if someone could give me an example of a group such that the following happens a $\equiv$ b (mod N) and c $\equiv$ d (mod N) but ac $\not\equiv$ bd (mod N) where N is a ...
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0answers
11 views

infinite intersection of jordan measurable sets

Is the infinite intersection of jordan measurable sets also jordan measurable? I´ve been trying to find a counterexample, but nothing so far. So is the statement true?
3
votes
1answer
90 views

Two categories sharing the same objects and morphisms

Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different ...
3
votes
1answer
62 views

Is the symmetric algebra direct sum of $k$-th symmetric powers?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $S(M)$ be the symmetric algebra of $M$ and $S^n(M)$ be $n$-th symmetric power of $M$. Then is $S(M)$ the internal direct sum of $S^n(M)$? ...
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1answer
56 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
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0answers
62 views

Examples for when differentiability fails

Let $l^1(\mathbb{N};\mathbb{R})$ be the set of all sequences $\mathbb{N}\to\mathbb{R}$ such that $\sum_{n\in\mathbb{N}}|x_n|<\infty$ for all $x\in l^1(\mathbb{N};\mathbb{R})$, together with the ...
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3answers
33 views

Sets with one accumulation point

Are there any more examples of sets in $\mathbb R$ that have one accumulation point apart from convergent sequences? I can´t think of any
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1answer
86 views

Extraneous solutions.

I just learned of extraneous solutions on the internet and thought, "could you both lose and gain solutions in one equation?" I think that, yes, you should be able to do that. However I haven't been ...
3
votes
2answers
75 views

Which optimization problem to use when I want to explain the concept of optimization to a layman?

I am looking some problem, which would be: Easy to understand Hard to solve intuitively Touch our everyday lives I am doing research in optimization using evolutionary computation. When people ...
0
votes
1answer
41 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
3
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0answers
76 views

understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
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1answer
49 views

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don't understand it. The ...
3
votes
1answer
69 views

What is an example of $R\otimes_R M$ not isomorphic to $M$?

Let $R$ be a ring and $f:R\rightarrow R$ be a ring homomorphism. ($f(1)=1$) Let $R$ be given the left operation as the ring operation on $R$ and the right operation as $x•r=xf(r)$, so that $R$ is an ...
7
votes
4answers
92 views

Example for $\dfrac{p_1p_2-1}{p_1+p_2}$ being odd natural number .

If $p_1,p_2$ are odd prime numbers , is it possible that $\dfrac{p_1p_2-1}{p_1+p_2}$ is odd natural number greater than 1.
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0answers
32 views

Similar function to Sinc function?

I am interested in a function which at $x=0$ starts from approximately $1$ and as you go on it decreases periodically to $0$ in a similar fashion to the Sinc function.
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votes
5answers
3k views

An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
0
votes
1answer
29 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
3
votes
1answer
37 views

Examples for almost-semirings without absorbing zero

What is an instructive example of a set $X$ equipped with two monoid structures $(X,+,0)$, $(X,\cdot,1)$, such that $+$ is commutative, the distributive laws hold, but $0 \cdot x = 0$ or $x \cdot 0 = ...
2
votes
0answers
55 views

A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
12
votes
2answers
1k views

What is the intuition between 1-cocycles (group cohomology)?

This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles? Let me explain a bit. Given an action of a group $G$ on another ...
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vote
0answers
90 views

P = NP, NP example problems in our daily life

For a little presentation for school, i want to try to explain the P=NP? Problem. I'm searching for examples for daily life NP-problems. (example: is making the weather forecast a NP problem?) And if ...
0
votes
1answer
37 views

Infinite sum of random variables is infinite

I am trying to better understand this statement and the assumptions made: If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, ...
3
votes
1answer
56 views

Example of equivalence relation where reflexivity/symmetry is not trivial

For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some ...
12
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6answers
800 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
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1answer
44 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
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2answers
118 views

Examples of Separable Spaces that are not Second-Countable

In this post I give an example of a separable space that is not second-countable. What are other good examples?
2
votes
1answer
349 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
0
votes
5answers
974 views

Counter-examples of homeomorphism

Briefly speaking, we know that a map $f$ between 2 topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous. So, can anyone give me 2 counter ...
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1answer
12 views

What is an example of a module over a division ring with two different ranks?

Let $R$ be a division ring and $M$ be an $R$-module. What is an example of $M$ and two bases $A,B$ of $M$ such that $|A|\neq |B|$?
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1answer
86 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...