# Tagged Questions

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 ...

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 ...
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 ...
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 ...
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 ...
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 ...
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 $$\frac{64}{16}.$$ ...
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], ...
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 ...
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} ...
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 ...
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 ...
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: ...
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 $\sum_n_=_1^\infty a_n$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 $f:\mathbb{R}^2\to\mathbb{R}$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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!
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 ...
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 ...
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 ...
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 ...
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 ...
2answers
153 views

### Noncompact sequentially compact space

Have you an example of a noncompact sequentially compact space, without using ordinal?
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 $\lambda>0$ ...
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 ...
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 ...
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 ...
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 ...
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} ...
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 ...
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 ...
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 ...
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 ...
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 $$g(x)=e^{-1/x^2}$$ ...
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