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)

0
votes
1answer
78 views

What is a counterexample that a composition of covering maps not a covering map?

Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...
4
votes
2answers
43 views

Example of a non-polynomial function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x)$ is negative for $x<0$ and positive for $x \ge 0$.

I have a bunch of polynomial functions example easily (e.g. $x^2$), but have trouble coming up with a non-polynomial function. I was thinking of defining $f(x) = e^{-x}$ for $x<0$ and $f(x) = ...
2
votes
2answers
30 views

Double complex with exact rows

Let $(K^{p,q},\delta,d)$ be a double complex of modules. We assume that $\delta$ of degree $(1,0)$, $d$ has degree $(0,1)$ and $d$ and $\delta$ commute. Since $d$ and $\delta$ commute, then $\delta$ ...
1
vote
1answer
36 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
7
votes
1answer
92 views

$X$ is connected, $A \subset X$ connected, and $C$ a component of $X\backslash A$. Is $\overline A \cap \overline C \ne \emptyset$?

I'm trying to prove or disprove the following statement: If $X$ is connected, $A \subset X$ is connected, and $C$ a component of $X\backslash A$ then $\overline A \cap \overline C \ne \emptyset$. I ...
4
votes
1answer
56 views

Continuous and bounded imply uniform continuity?

I am thinking about this since couple hour. Is a continuous and bounded function $f:\mathbb R\to\mathbb R$ uniform continuous too? I didn't found a counter example and thus I tried to prove this like ...
3
votes
1answer
68 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
1
vote
1answer
33 views

Can this statement about winding number generalized?

Definition Let $\alpha$ be a path in $\mathbb{C}\setminus\{z_0\}$. Since $\mathbb{C}\rightarrow \mathbb{C}\setminus \{z_0\}:z\mapsto e^z$ is a covering map, $\alpha$ can be decomposed as ...
0
votes
0answers
19 views

What is an example that Euclidean function varies by a unit?

Let $R$ be a Euclidean domain and $f$ be a Euclidean function on $R$. (Not necessarily submultiplicative) Let $a,b$ be nonzero elements of $R$ such that $a=ub$ where $u$ is a unit of $R$. Is there ...
5
votes
1answer
55 views

Example of a surjective local homeomorphism that is not a covering? [duplicate]

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering? I don't think this ...
0
votes
2answers
28 views

Continuous marginal distributions do not imply continuous joint distribution

I already proved the other implication. I need to find an explicit example that shows that if there is some random vector $(X,Y)$ and $X$ and $Y$ have both continuous marginal distributions, then ...
2
votes
2answers
39 views

Existence of an metric or a topology so that every subset is compact

Let $X$ be a infinite set. Is there a metric on $X$ such that every sub set of $X$ is compact? What about a topology on $X$? I think that if we can answer first question then we can answer the ...
4
votes
0answers
42 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
1
vote
1answer
67 views

Sobolev functions counterexample

Let $A=(0,1)^{d}$.Does anyone have a simple example of a funtion in $H_0^1(A)\cap H^2(A)$ that is not in $H^2_0(A)$? Thanks a lot.
14
votes
1answer
201 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
0
votes
0answers
23 views

Examples of ODEs with a 3-dimensional vector-function

I'm testing a program which solves ODEs and I need some examples. Where can I find some examples of ODEs with a 3-dimensional vector-funcion? Likes these two:
2
votes
3answers
99 views

Continuous bijective function between the same topology that is not a homeomorphism.

I know there are many examples when the domain and co-domain do not coincide. Taking the identity on $X$ from $(X,\tau_1)$ to $(X,\tau_2)$ when $\tau_2$ is coarser than $\tau_1$ gives an infinite ...
0
votes
0answers
75 views

non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
0
votes
0answers
52 views

Is there an example such that $\text{rank}(A^t)\neq \text{rank}(A)$?

Let $R$ be a commutative Noetherian ring and $A\in M_{n\times m}(R)$. If $R$ is a field, then $rank(A^t)=rank(A)$. However, in general Noetherian rings, does $rank(A^t)=rank(A)$ hold? Since the ...
6
votes
0answers
61 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?
0
votes
1answer
17 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 ...
1
vote
3answers
55 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
votes
1answer
29 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 ...
2
votes
0answers
83 views
+100

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 ...
2
votes
0answers
53 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
357 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 ...
3
votes
1answer
65 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 ...
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 $\alpha \in V$ ...
3
votes
1answer
79 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} ...
2
votes
3answers
123 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 .
1
vote
2answers
48 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
0
votes
1answer
28 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 ...
2
votes
0answers
26 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 ...
1
vote
1answer
33 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 ...
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 ...
3
votes
2answers
93 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 ...
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) ...
2
votes
1answer
41 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 ...
2
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 $[0,\infty)$. ...
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
39 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
46 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
68 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 ...
0
votes
3answers
32 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
44 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: ...
2
votes
0answers
24 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 ...
0
votes
1answer
51 views

Differentiability at a point does not imply differentiability on a neighbourhood. Do all similar implications fail?

(Write $[]$ for the Iverson bracket.) I recently learned about the function $$\mathbb{R} \rightarrow \mathbb{R}$$ $$x \mapsto [x \in \mathbb{Q}]x^2 +[x \notin \mathbb{Q}](-x^2)$$ which is ...