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)

10
votes
0answers
159 views

Hopf-like monoid in $(\Bbb{Set}, \times)$

I am looking for a nontrivial example of the following: Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), ...
1
vote
1answer
59 views

Is the intersection of two uniformities a uniformity?

Is there any two uniformities $\mathcal{D}_1$ and $\mathcal{D}_2$ on a set $X$ such that $$\mathcal{D}_1\cap \mathcal{D}_2$$ is not a uniformity on $X$?
1
vote
1answer
80 views

Suppose every complete subset of a uniform space is closed. Is it Hausdorff?

Suppose $(X,\mathcal{D})$ is a uniform space. for each nonempty subset $A\subseteq X$, if the subspace $(A,\mathcal{D}_A)$ is complete then $A$ is closed. Is $(X,\mathcal{D})$ Hausdorff?
8
votes
1answer
206 views

Categorify a proof?

I am quite new to categories and the book I am reading is Lawvere and Schanuel's Conceptual Mathematics. At the end of Part 2 the authors use the proof of Brouwer's fixed point theorems as an ...
1
vote
1answer
134 views

Is there a first countable and star countable space which is not separable?

A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. My ...
1
vote
2answers
174 views

Is the topology generated by the open intervals of a partially ordered set necessarily T_0?

Let $X$ denote a set and consider a collection $C \subseteq \mathcal{P}X$. Let $\mathcal{T}$ denote the set of all topologies $T$ on $X$ such that $C \subseteq T$. Then $\bigcap \mathcal{T}$ is a ...
1
vote
0answers
36 views

Is there a nonvirtually abelian group whose commutator subgroup is finite? [duplicate]

Possible Duplicate: If the derived subgroup is finite, does the center have finite index? Let $G$ be a finitely generated group whose commutator subgroup is finite. Is $G$ necessarily ...
4
votes
2answers
475 views

Looking for a bijective nowhere-continuous function ${\mathbb R}\rightarrow{\mathbb R}$

Does there exist a bijective function $f:{\mathbb R}\rightarrow{\mathbb R}$ that is nowhere-continuous, assuming that both domain and range have the "standard topology"? 1 1 By this I mean the ...
6
votes
1answer
604 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
3
votes
1answer
116 views

What are the use cases related to cluster analysis of different distance metrics?

I'm trying to use different distance metrics like Euclidean, Manhattan, cosine, chebyshev among other distance metrics in my k-means algorithm to calculate distances between the data points and the ...
16
votes
5answers
295 views

Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples

In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
0
votes
1answer
24 views

On the existence of functions with a particular convergence

Is the following scenario possible? Provide an example or argue why not. Let $\{f_n\}_{n=1}^{\infty}$ be measurable non-negative functions on $[0,1]$ converging to $f(x)$ pointwise Lebesgue-almsot ...
1
vote
1answer
86 views

Looking for a counterexample [duplicate]

Possible Duplicate: Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) I am looking for a function $f:\mathbb{R}\rightarrow \mathbb{R}$ that for all $x$ and $y$ ...
87
votes
1answer
5k views

How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many ...
3
votes
0answers
58 views

Differentiable function which is nowhere continuously differentiable [duplicate]

Possible Duplicate: How discontinuous can a derivative be? $x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$. But is there ...
0
votes
2answers
315 views

Vector field with bounded integral curves

I am thinking about smooth vector fields on some (open set of an) euclidean space $\mathbb{R}^n$. I know that the integral curves of a general vector field $X$ are not defined for every time $t\in ...
0
votes
1answer
58 views

for positive functions $f(n)$ and $g(n)$, can $f(n)$ be in $\mathcal{O}(g(n))$ and $\Omega(g(n))$?

For positive functions, is it possible for $f(n)$ to be lower bounded by $g(n)$ if its already being upperbounded by $g(n)$? If $f(n) = g(n) = n$, then doesn't that mean $g(n)$ is a lower and ...
4
votes
2answers
435 views

Draw a non-planar graph whose complement is a non-planar graph

I have been teaching myself graph theory. I am stuck at solving this problem on my own. Please provide an example of such a graph. What approach would you take to draw such a graph?
3
votes
1answer
115 views

How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...
1
vote
0answers
56 views

Non context-free languages closed under reverse

Is this sentence true or false? I'll be glad for some explaination L is not context-free, then its reverse is also not context-free Thanks in advance
6
votes
1answer
142 views

Example of a pair of non-cobordant manifolds

So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not ...
4
votes
0answers
80 views

Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
1
vote
3answers
2k views

5 linear equations in 5 unknowns

I need an example of 5 linearly independent equations with 5 variables. How can I write such a equation set. As an example: ...
9
votes
1answer
181 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
2
votes
2answers
313 views

Let $G$ be a group of order $56$. Then which of the following are true

Let $G$ be a group of order $56$. Then which of the following are true All $7$-sylow subgroups of $G$ are normal All $2$-Sylow Subgroups of $G$ are normal Either a $7$-Sylow subgroup or a ...
4
votes
1answer
49 views

Questions about an example

Recently, I met an example. I have two questions about the example: Why the author said, because $z \notin A$, then $z$ is not in the closure in $\beta \mathbb{R}$ of $A \cap (\beta \mathbb{R} ...
11
votes
3answers
346 views

Is every contractible space a cone?

It is easy to show that for any topological space $X$, the cone $CX$ is contractible. I am interested in the converse. If $Y$ is a contractible space, is $Y$ homeomorphic to $CX$ for some topological ...
10
votes
4answers
227 views

Examples of Monads and their Algebras

I'd like to get some examples of monads; specifically, I'd love a big list of different monads and a description of what their algebras are. Alternatively, online resources and especially exercices ...
4
votes
2answers
225 views

Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.

How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?
1
vote
0answers
49 views

Duality between $[G,G]$ and $Z(G)$? [duplicate]

Possible Duplicate: Center-commutator duality Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation} ...
9
votes
1answer
365 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
8
votes
2answers
2k views

Compactly supported function whose Fourier transform decays exponentially?

It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
1
vote
1answer
153 views

On finite groups whose center is elementary abelian group

Let $G$ be a finite 2-group such that $Z(G)$ is elementary abelian 2-group ($\mid Z(G)\mid\geq 4$) and $Inn(G)$ is of order 4. Then prove that there exists an $\alpha\in Aut(G)$ such that ...
1
vote
1answer
265 views

Example of a non-injective retract induced homomorphism of fundamental groups

When answering this question I used the fact that when we have a retract $r:X \rightarrow Y$ the induced homomorphism $r_\ast: \pi_1(X) \rightarrow \pi_1(Y)$ is surjective. I can recall how to prove ...
12
votes
2answers
1k views

Uniqueness of product measure (non $\sigma$-finite case)

Let $(X,\mathscr{A},\mu), (Y,\mathscr{B},\nu)$ be two measure spaces, then we have the product measurable space $(X\times Y, \mathscr{A}\times\mathscr{B})$ where $\mathscr{A}\times\mathscr{B}$ is the ...
5
votes
3answers
320 views

If $|f(x)|$ is a differentiable function, then $f(x)$ is also?

If $|f(x)|$ is a differentiable function, then $f(x)$ is also a differentiable function. Why is this wrong? Can you find a counterexample please? It seems like a true sentence.
6
votes
2answers
336 views

Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}Q\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
4
votes
5answers
159 views

irrationality of numbers with rational sum

Assume that $x_1, \dots, x_n$ are non-negative real numbers such that $$ x_1 + \dots + x_n \in \mathbb Q~~~~~~~~~~~~~~ \text{ and } ~~~~~~~~~~~~~~~x_1 + 2x_2 + \dots + nx_n\in \mathbb Q. $$ Does ...
16
votes
6answers
2k views

Uncountable closed set of irrational numbers

Could you construct an actual example of a uncountable set of irrational numbers that is closed (in the topological sense)? I can find countable examples that are closed, like $\{ \sqrt{2} + ...
8
votes
2answers
251 views

What was Klein working on when he “replaces his Riemann surface by a metallic surface”?

I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting: Look at Professor Klein: he is studying one of the most abstract questions of ...
12
votes
1answer
287 views

An undergraduate level example where the set of commutators is proper in the derived subgroup.

The derived subgroup is the subgroup generated by the set of all commutators of a group $G$. I always used to forget that "generated by" part. Soon I will be teaching a group theory course and wish ...
2
votes
1answer
89 views

Any example that $f_n\rightarrow f$pointwise and $f_n'\rightarrow f'$uniformly, but not $f_n\rightarrow f$uniformly?

Let $C$ be an infinite connected set in $\mathbb{R}$ and $\{f_n\}$ be a sequence of differentiable functions from $C$ to $\mathbb{R}^k$. Suppose (i)$f_n'$ coverges uniformly $//$ (ii)There exists ...
3
votes
1answer
223 views

Shortest triangulation is in general not a Delaunay triangulation

Let $P$ be a set of points. The minimal triangulation of $P$ is a triangulation $T$ of the points in $P$ such that the total length of the edges in $T$ is the smallest possible amongst all possible ...
6
votes
2answers
176 views

Is it possible that $H\cap g^{-1}Hg$ is a nontrivial proper subgroup of $H$?

Given a group $G$ and two conjugated subgroups $ H $ and $ H'=gHg^{-1} $, is the following proposition true? There are only two possibilities for the subgroups: either $ H\cap H' = 1 $ or $ H=H'$. I ...
1
vote
1answer
43 views

Relations between a product in $L^p$ and essential boundness of a factor

Let be $1\leq p<\infty$ and $g$ a measurable funtion defined on $E$. I have to prove that if $fg\in L^p$ for every $f\in L^p(E)$, then $g$ is essentialy bounded, that is $g\in L^\infty (E)$. I ...
1
vote
1answer
208 views

Uniform convergence and complete metric space

Let $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$. Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} ...
7
votes
2answers
2k views

What is an example that a function is differentiable but derivative is not Riemann integrable

I have two questions that i'm curious about. If $f$ is differentiable real function on its domain, then $f'$ is Riemann integrable. If $g$ is a real function with intermediate value property, then ...
2
votes
1answer
121 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 ...
3
votes
1answer
112 views

A countale partially ordered set that has an uncountable number of maximal chains

I'm looking for a countable set S with a partial order < that has an uncoubtable number of maximal chains. I had many ideas but non of then is correct (for example- S= natural numbers, "<" is ...
25
votes
17answers
1k views

Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...