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 ...
10
votes
1answer
169 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 ...
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 ...
4
votes
5answers
144 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 ...
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 ...
101
votes
23answers
5k views
Can't argue with success? Looking for “bad math” that “gets away with it”
I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare").
One example would be "cancelling" the 6s in
$$\frac{64}{16}.$$
...
3
votes
0answers
54 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
340 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 ...
1
vote
2answers
119 views
An epimorphism in $\text{Grp}$ without right inverse?
Exercise 8.24 in Aluffi's Algebra: Chapter 0 asks us to find an epimorphism in $\text{Grp}$ without right inverses.
I happen to know that epimorphisms in $\text{Grp}$ are surjective, so we need a ...
1
vote
3answers
137 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:
...
0
votes
2answers
115 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 $\sum_n_=_1^\infty a_n$ ...
11
votes
4answers
271 views
Counterexamples in complex analysis
In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ...
14
votes
2answers
281 views
Is every group the automorphism group of a group?
Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$?
EDIT: It has been pointed out that the answer ...
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 ...
1
vote
2answers
100 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 ...
2
votes
1answer
89 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 ...
3
votes
1answer
68 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 ...
1
vote
1answer
85 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
149 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.
1
vote
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 ...
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 $f:\mathbb{R}^2\to\mathbb{R}$ ...
2
votes
1answer
164 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 ...
18
votes
16answers
2k views
An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.
In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English ...
8
votes
2answers
135 views
Almost A Vector Bundle
I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing ...
2
votes
1answer
43 views
So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?
For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation}
H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\},
\end{equation} where $\mathcal{S}'$ is the space of ...
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 ...
2
votes
3answers
148 views
Example of two-dimensional non-abelian Lie algebra?
can some one give me an example of two-dimensional non-abelian Lie algebra?
0
votes
3answers
296 views
iid variables, do they need to have the same mean and variance?
If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different ...
1
vote
1answer
64 views
Sequence of continuous fuctions $f_n:[0,1]\rightarrow [0,1]$ s.t. $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ but…
Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but ...
0
votes
2answers
56 views
Counterexample of Existence of a continuous extension of a Continuous function
Till now, I have proved followings;
Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,
$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous ...
2
votes
3answers
441 views
Is a bounded and continuous function uniformly continuous?
$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous?
Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!
2
votes
0answers
88 views
Ways to look at categories [closed]
The axioms of category theory arise in many contexts
structures and maps between them
structures and interpretations between them
theories and interpretations between them
theories and proofs ...
2
votes
3answers
156 views
Analytic Function with positive integers as zeros?
Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ?
If yes, please give some example.
PS: I already thought of ...
7
votes
1answer
159 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
66 views
How to show that linear span in $C[0,1]$ need not be closed [duplicate]
Possible Duplicate:
Non-closed subspace of a Banach space
Let $X$ be an infinite dimensional normed space over $\mathbb{R}$. I want to find a set of vectors $(x_k)$ such that the linear ...
3
votes
2answers
454 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 ...
5
votes
2answers
153 views
Noncompact sequentially compact space
Have you an example of a noncompact sequentially compact space, without using ordinal?
6
votes
2answers
68 views
Alternate definition for boundedness in a TVS
Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if
for any open neighborhood $N$ of $0$ there is a number $\lambda>0$
...
0
votes
2answers
104 views
How to cook up test functions?
Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But ...
3
votes
1answer
370 views
Poset Infimum and Supremum
I was asked to show that if every subset of a poset has an infimum then every such subset has a supremum. I did my proof and now I realize that what I was calling "infimum" was actually "a smallest ...
4
votes
1answer
114 views
Is this AM/GM refinement correct or not?
In Chap 1.22 of their book Mathematical Inequalities, Cerone and Dragomir prove the following interesting inequality. Let $A_n(p,x)$ and $G_n(p,x)$ denote resp. the weighted arithmetic and the ...
6
votes
1answer
344 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 ...
1
vote
1answer
51 views
Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?
Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation}
...
0
votes
2answers
138 views
Providing a counter example for a Logic Statement
How do I give a counter-example of the following logic statement (I think the statement is false):
There exists $x$ $\geq$ 0 s.t. (For All real $y$, $x$ = $y$$^2$)
Since the statement has a "There ...
4
votes
4answers
186 views
Examples of partial functions outside recursive function theory?
My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.
In recursive function theory one studies partial functions on the set of natural numbers.
Are ...
13
votes
4answers
569 views
Is a vector space over a finite field always finite?
Definition of a vector space:
Let $V$ be a set and $(\mathbb{K}, +, \cdot)$ a field.
$V$ is called a vector space over the field $\mathbb{K}$ if:
V1: $(V, +)$ is a commutative group
V2: $\forall ...
3
votes
2answers
108 views
Question about the cardinality of a space
I've been having conflicting thoughts about the following problem, and I was wondering if anyone could help me out.
Is is true that the cardinality of every regular separable space does not ...
4
votes
2answers
209 views
nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$
When I teach second-semester calculus I usually discuss the function $f$ defined by
$$ f(x)=e^{-1/x^2} $$
for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by
$$ g(x)=e^{-1/x^2} $$
...
