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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2
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1answer
308 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
5
votes
0answers
26 views

Does there exist a graph $G$ such that every edge is contained in a unique Hamiltonian circuit, that is not a cycle graph?

Suppose $G$ is an (undirected, simple) finite graph. If $G$ is a cycle graph, then each edge of $G$ belongs to a unique Hamiltonian circuit. Does there exist a non-cycle graph $G$ with this property?
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0answers
23 views

Approximate theorems

There was a popular discussion a while ago on proofs only found with large counterxamples: Examples of apparent patterns that eventually fail. I was think the converse might be interesting. I'd like ...
0
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1answer
78 views

Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
0
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1answer
15 views

Is my example of non equivalent maps correct?

We define two smooth maps $f: (\mathbb R, 0) \to (\mathbb R^2, 0)$ and $g: (\mathbb R, 0) \to (\mathbb R^2, 0)$ to be equivalent if there exist diffeomorphisms $\tau : \mathbb R \to \mathbb R$ and ...
3
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1answer
63 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
3
votes
2answers
137 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
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3answers
53 views

Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. ...
2
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1answer
28 views

Counterexample to a variation on “The politician theorem”.

The following is a theorem in graph theory that has a nice 'real world' interpretation: Suppose $G$ is a finite simple graph in which any two vertices have precisely one common neighbour. Then ...
3
votes
2answers
438 views

K topology: Examples

Why would the interval $(-3,1)$ be open in the $k$-topology? (I'm using Munkres). Can I have some other examples of intervals in $k$-topology? What exactly does $(a,b)$ $\cup$ $(a,b)-k$ for ...
74
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32answers
7k views

What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...
5
votes
3answers
845 views

A locally injective but non globally injective function?

A continuous function $f : U \subset \mathbb{R}^n \to \mathbb{R}^n$, is said to be locally injective at $x_0 \in U$ if exist a neighborhood $V \subset U$ of $x_0$ s.t. $f|_V$ is injective. $f$ is said ...
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0answers
41 views
+50

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
4
votes
1answer
65 views

Non-example for vector space

$V$ is a vector space over a field $F$ if it satisfies the following conditions. $(V,+)$ is an abelian group. $1 \in F $ such that $1.\alpha=\alpha$ for every $\alpha \in V$ ...
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2answers
254 views

Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not ...
2
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0answers
48 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
5
votes
2answers
351 views

An example for infinite dimensional vector space with Hamel dimension smaller than $\operatorname{card} F$

What will be the example for a vector space(infinite dimensional) over a field where Hamel basis has strictly smaller cardinality than that of field? It is not possible in a Hilbert Space (over R or ...
9
votes
2answers
1k views

find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...
3
votes
1answer
77 views

Counter exchanging limit and integral

Background I came across this answer on Math SE which claimed it made a lot of sense to switch limit and integral. In response I came up with the following counter-examples: $\lim_{w \to 0} ...
61
votes
0answers
1k 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, $f(S)\subset Y$ ...
2
votes
3answers
90 views

Looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points

I am looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points ; please help , thanks in advance .
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2answers
46 views

If $\sum a_{2k}$ exists then $\sum a_m$ exists?

Let $\{a_0,a_1,a_2...\}$ be a sequence of real numbers let $s_n=\sum a_{2k}$. If $\lim_{n\rightarrow \infty} s_n $ exists then $\sum a_m$ exists. Is it true? I don§t find a counter example
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votes
0answers
70 views

Finding a weight function with a specific property

I am looking for a (smooth, quickly decaying) function $w : [0,\infty) \rightarrow \mathbb{R}$ such that $$w(t) \cdot \int_{0}^{t} \frac{1}{w(2x)} dx $$ is absolutely integrable on $[0,\infty)$. ...
0
votes
1answer
27 views

If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments ...
6
votes
1answer
156 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
2
votes
0answers
21 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
0
votes
1answer
23 views

A simple example of a base point of a linear series

I'm reading Fulton's algebraic curves book and he make the following definition of linear series (page 110): Let $D$ be a divisor, and let $V$ be a subspace of $L(D)$ (as a vector space). The ...
2
votes
1answer
27 views

Convergence in distribution with finite mean

I'm preparing myself for the final exam of my graduate Probability Theory course and was stuck with another one of the exercises our professor gave us. Let $X_n, n=1,2,\ldots,$ and $X$ be nonnegative ...
1
vote
1answer
32 views

What's an example where this definition of measure has a limit that does not exist?

I'm looking for an example. Consider defining the measure of a set $E \subseteq \Bbb{R}^d$ by a limit: Take $$m(E) := \lim_{N \to \infty} \frac{1}{N^d} \cdot \left| E \cap \frac{1}{N} \Bbb{Z}^d ...
3
votes
2answers
91 views

Does there exist a sequence $\{a_{n}\}$, but $\lim_{n\to\infty}a_{n}\neq 0$

Does there exist a sequence $\{a_{n}\}$, such that $$\lim_{n\to\infty}(a_{n+1}-a_{n})=0\ \ \ \text{and} \ \ \lim_{n\to\infty}\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}=0$$ but $$\lim_{n\to\infty}a_{n}\neq 0 ...
18
votes
1answer
386 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
0
votes
0answers
16 views

example of a quadratic form

Would someone be able to tell me an example of a quadratic form defined on a five-dimensional vector space $V$ over a non-archimedean local field $k$ of positive characteristic (not equal to two, say) ...
1
vote
2answers
99 views

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

Does there exist an algebraic structure $(\mathbb{K},+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in \mathbb{K}$?
2
votes
1answer
32 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
3
votes
4answers
722 views

Example for a proper dense subspace?

I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But ...
5
votes
1answer
45 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
0
votes
0answers
22 views

More on the implicit function theorem: is this example correct?

I am trying to understand the implicit function theorem so I thought it would be a good idea to work out an example. Please could someone look at this and tell me if it is correct? Consider the ...
0
votes
2answers
34 views

Set theory basics exam

I have a question about this, we had this on our exam. Let be $f:A \to B$ a function. Prove next statements or give an example against it. (i): if $A$ is countable, then $f(A)$ is also ...
2
votes
3answers
41 views

Interesting examples of non-normal operators?

I am currently learning spectral aspects of linear algebra. At first sight, it seems like normality is very narrow restriction. But, I can not think up any examples of non-normal operators. There is ...
4
votes
1answer
246 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \dfrac{x_1 + \cdots+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \dfrac{x_1 + ...
4
votes
1answer
67 views

Counter Example of Continuous Functions

I came across this question: Question: Let $f$ be a real continuous function defined on [0,1] such that $f(0) = 0$ and $f(1) = 1$. Prove or give a counter-example to the following: a) If $f'$ ...
4
votes
2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
2
votes
0answers
61 views

Counter-example to exponential law for locally compact [non-Hausdorff] spaces

There is a natural bijection $$ \operatorname{Map}(X\times Y,Z)\cong\operatorname{Map}(X,\operatorname{Map}(Y,Z)),\quad f\mapsto(x\mapsto(f(x,{-})). $$ If $X$, $Y$, $Z$ are topological spaces one can ...
1
vote
0answers
47 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
0
votes
3answers
26 views

A semigroup with three or four elements without identity [duplicate]

Does there exist a semigroup with three or four (or finite) elements, without identity? I tried to construct such an example, but every example I tried to construct had an identity element.
4
votes
1answer
39 views

Anomalous finitary objects

I think it was about 15 years ago that Igor Pak told me that the symmetric group on six elements has outer automorphisms, and I was startled. Somehow that had escaped my notice. That one is a freak: ...
1
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0answers
28 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
8
votes
3answers
249 views

Is every monoid isomorphic to its opposite

This may be a trivial question. Every group is isomorphic to its opposite using the isomorphism that sends $x$ to $x^{-1}$. Now does this hold even if the condition of existence of inverses is ...
2
votes
0answers
23 views

Monomorphisms into direct products

Let $G$ be a group. I am interested in the following property: For any groups $A,B$ and monomorphism $G \hookrightarrow A \times B$, either $G \hookrightarrow A$ or $G \hookrightarrow B$. For ...
3
votes
3answers
139 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 $[0,1]$ ?