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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2answers
128 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
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1answer
46 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
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2answers
170 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?
8
votes
4answers
959 views
Famous uses of the inclusion-exclusion principle?
The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that ...
3
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1answer
77 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 ...
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0answers
33 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
5
votes
1answer
51 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
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0answers
51 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
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3answers
140 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:
...
8
votes
1answer
140 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 ...
1
vote
2answers
152 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 ...
5
votes
1answer
44 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} ...
9
votes
3answers
196 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 ...
8
votes
4answers
120 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
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2answers
127 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
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0answers
42 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}
...
7
votes
1answer
303 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?
7
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2answers
779 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
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1answer
115 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
62 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 ...
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0answers
37 views
Example ideal of $\mathfrak{sl}(2,\mathbb{C})$
I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me?
I try to make ideal except trivial ...
10
votes
2answers
751 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 ...
2
votes
1answer
108 views
On groups whose center has odd order
Let $G$ be a finite group such that $Z(G)$ is of odd order and $Inn(G)$ is of even order. Then prove $G\simeq Z(G)\times N$, such that $N$ is a subgroup of $G$ where $N\simeq Inn(G)$. Thank you
5
votes
3answers
259 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.
5
votes
2answers
140 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
145 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 ...
17
votes
6answers
816 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} + ...
3
votes
2answers
464 views
Differential Equations without Analytical Solutions
In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is:
Can you prove a differential equation ...
8
votes
2answers
147 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 ...
10
votes
1answer
170 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
70 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 ...
2
votes
1answer
40 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
123 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 ...
3
votes
0answers
56 views
Application of a result on some bounded functionals on a subspace of $C([0,1])$
The following result was proved in a previous post:
Bounded functionals on Banach spaces.
Let $(X, \|.\|)$ be a Banach space such that
$X \subset C([0,1]) $
For every $r\in \mathbb{Q}\cap[0,1], ...
1
vote
1answer
29 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 ...
0
votes
1answer
87 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} ...
3
votes
2answers
351 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 ...
0
votes
2answers
117 views
Weierstrass M-test proof?
Let (X,d) be a metric space. For each n $\epsilon$ N let $g_n$:X$\rightarrow$R be a continuous function. Let ($a_n$) be a sequence of positive real numbers such that the series $\sum_n_=_1^\infty a_n$ ...
2
votes
1answer
90 views
Applications of the Pontryagin product for abelian groups
For an abelian group $G$, one can give an explicit description of the homology ring $H_*(G, k)$ for e.g. $k=\mathbb{Q}, \mathbb{Z}_p$ or in general PIDs $k$ in which every natural number is ...
2
votes
1answer
86 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 ...
1
vote
2answers
102 views
Does the Laplace transform biject?
Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.'
Can someone provide a proof or counterexample ...
3
votes
1answer
69 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 ...
20
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17answers
988 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 ...
2
votes
1answer
165 views
Non-physical Jounce Examples in Nature
What are some good examples of jounce in the non-physics arena?
The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena and B) you never hear of too many ...
2
votes
2answers
206 views
How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?
I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
13
votes
4answers
578 views
Is a vector space over a finite field always finite?
Definition of a vector space:
Let $V$ be a set and $(\mathbb{K}, +, \cdot)$ a field.
$V$ is called a vector space over the field $\mathbb{K}$ if:
V1: $(V, +)$ is a commutative group
V2: $\forall ...
1
vote
1answer
86 views
What would be a counterexample if hypothesis of mean value theorem is slightly changed?
Let $f:[a,b] \rightarrow \mathbb{R}$ be a function which is continuous on $(a,b]$ and differentiable on $(a,b)$. Is there any function such that $f(b)-f(a)≠(b-a)f'(x), \forall x\in (a,b)$?
There was ...
2
votes
3answers
152 views
Locally Compact Hausdorff Space That is Not Normal
Someone told me that locally compact Hausdorff spaces (unlike compact ones) need not be normal. Can one give me please such an example? Thank you.
3
votes
1answer
217 views
Continuous partials at a point but not differentiable there?
In Question on differentiability at a point, it is mentioned (and in Equivalent condition for differentiability on partial derivatives it is cited from Apostol) that for $f:\mathbb{R}^2\to\mathbb{R}$ ...
1
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0answers
58 views
Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
