4
votes
2answers
133 views

homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must ...
3
votes
1answer
75 views

Weyl Operator Group Actions

For $g \in L^2(\mathcal{R})$ and real numbers $p$ and $q$, denote $g^{(p,q)}(t) = e^{ipt}g(t-q)$. Calculate $||g^{(p,q)}||, M_t(g^{(p,q)}), M_{\omega}(g^{(p,q)}), \sigma_t(g^{(p,q)}),$ and ...
4
votes
0answers
46 views

Is there a general theory of closed-form expressions?

Many expressions with an infinite number of terms can be rewritten in closed form using a certain set of "elementary" functions, and many of those relations are not at all obvious at first sight, for ...
9
votes
3answers
124 views

Can $\mathbb{R}$ be written as an ascending union of proper additive subgroups?

Can the group $\mathbb{R}$ be written as countable ascending union of proper subgroups? (i.e. does there exists a series of proper subgroups $H_1\leq H_2\leq \cdots $ such that $\cup ...
3
votes
2answers
150 views

Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
5
votes
0answers
127 views

Dense uncountable proper subgroup of $(\mathbb{R},+)$

Probably someone had asked this question on StackExchange, but can one construct a dense uncountable proper subgroup of $(\mathbb{R},+)$?
5
votes
2answers
186 views

Group theory with analysis

I've studied group theory upto isomorphism. Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems. I too have done metric spaces and real analysis properly. ...
2
votes
2answers
121 views

What is the difference between the words chord, tangent in (a) and (b)?

(a) If a function $g$ is continuous on the closed interval $[u,v]$, where $u<v$, and differentiable on the open interval $(u,v)$, then there exists a point $c$ in $(u,v)$ such that ...
5
votes
2answers
205 views

Primes and the Unit circle.

Consider the "prime spiral" $f(z) = \sqrt{z}\exp(2\pi i \sqrt{z})$, for integer $z$. It has been shown that the intersections of $f$ with some quadratic curves contain a significantly disproportionate ...
6
votes
1answer
179 views

Do proper dense subgroups of the real numbers have uncountable index

Just what it says on the tin. Let $G$ be a dense subgroup of $\mathbb{R}$; assume that $G \neq \mathbb{R}$. I know that the index of $G$ in $\mathbb{R}$ has to be infinite (since any subgroup of ...
0
votes
1answer
75 views

E measurable with m(E) < $\infty$?

Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$. ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable. I told my ...
0
votes
1answer
54 views

Additive maps modulo $1$ - what do they look like?

Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$). A map $f:\ T \to T$ is ...
0
votes
2answers
115 views

How I can prove that there is a bijection between the set $A$ and $ℤ$?

Let $f:ℝ→ℝ$ be a real analytic function and not identically zero. Assume that $f$ has infinitely many negative zeros. Let us consider the set $A$ of points $(c_{k},0)$ where $c_{k}<0$ is a zero of ...
3
votes
1answer
60 views

Suppose $f: \Bbb R \to \Bbb R$, where $f$ is continuous on $(-\infty, 0)$ and on $(0, \infty)$. Show that $f$ is measurable?

Prove that: Suppose $f \colon \Bbb R \rightarrow \Bbb R $, where $f$ is continuous on $(-\infty, 0)$ and on $(0,\infty)$. Show that $f$ is measurable. Can someone please help me with this proof? ...
1
vote
3answers
118 views

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. Show that $f$ is measurable?

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. ($x$ < $x'$ $\implies $ $f(x)$ < $f(x')$). Show that $f$ is measurable. I am a self taught person and just started ...
4
votes
3answers
182 views

what are all the open subgroups of $(\mathbb{R},+)$

I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
3
votes
1answer
302 views

Isomorphism from $\mathbb{R}$ to $(-1,1)$

There are many bijective functions that map $\mathbb{R}$ to $(-1,1)$, in particular: $$f\left(x\right)=\frac{e^{2x}-1}{e^{2x}+1}$$ (Of course there are others, such as ...
4
votes
1answer
68 views

Question sort of related to Cayley's theorem

In group theory, we saw that if $G$ is a group over a set $X$, then we can embed $G$ into $S_X$, where $S_X$ is the group of permutations on $X$, i.e. there is an injective homomorphism $G ...
1
vote
2answers
237 views

Lie group homomorphism from $\mathbb{R}\rightarrow S^1$

I need to prove that every Lie group homomorphism from $\mathbb{R}\rightarrow S^1$ is of the form $x\mapsto e^{iax}$ for some $a\in\mathbb{R}$. Here is my attempt: As it is group homomorphism so it ...
7
votes
2answers
766 views

How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
2
votes
1answer
371 views

What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]

Possible Duplicate: Is there a name for such kind of function? I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
17
votes
4answers
417 views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
4
votes
2answers
285 views

Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-1}$ ...
6
votes
1answer
165 views

Existence of measure zero generating sets of additive real numbers.

Just getting around to posting thoughts I had regarding this question about the additive structure of the real numbers. I was interested in which sets generate $(\mathbb{R},+)$. First, is the ...
2
votes
2answers
91 views

construct a Lipschitz function

Let $P_n$ be group of all permutations of the set $\{1,\dots,n\}$. I have trouble with construction a Lipschitz function $f:P_n \longrightarrow R$. Any help would be appreciate. Thank you.