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Group action and semidirect product

Let $m,n \in \mathbb N$, $k$ a field, $X=(k^{n\times m},+)$, and we consider the groups $GL(n,k)$ and $GL(m,k)$. Let $K:=GL(n,k) \times GL(m,k)$. We define \begin{align*} K\times X &amp;\to X\\ ((

mathematics

$L_1\subset L_p$?

I am trying to check whether the implication $\forall p&gt;1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)&lt;\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue inte

mathematics

Is there a word with the meaning to mix "s" and "th" when pronounced?

I remember I have met a word which means people cannot distinguish "th" from "s" when they speak but now I can't recollect it. So I need some help here. Does anyone know that w

english-language

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu&lt;\infty$ exists. In

mathematics

Unique morphism from the additive group $\mathbb Q$ to $\mathbb Z$

I am trying to prove that the only group homormorphism from $\mathbb Q$ to $\mathbb Z$ is the trivial one but I couldn't Suppose there is $x \in \mathbb Q$ : $f(x)=z \neq 0$. We can write

mathematics

Implicit Function Theorem: Second Derivative

After applying the Implicit Function Theorem for $f:\mathbb{R}^3 \to \mathbb{R}$ at $(x_0,y_0,z_0)$ and $f_z \neq 0$ it is assured that there is a function $\phi: \mathbb{R}^2 \to \mathbb{R}$ such ...

mathematics

Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.

Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented. Let the deRham cohomology class of $A$ be denoted $[\phi_A]$. The class is defined by

mathematics

Does limit of $x_n=(1+\frac{1}{2})*(1+\frac{1}{4})*...*(1+\frac{1}{2^n})$ exist?

I tried to show that the sequence is increasing and limited, but couldn't find the limit. I also tried with squeeze theorem, but $(1+\frac{1}{2^n})^n&lt;=x_n&lt;=(1+\frac{1}{2})^n$ is not

mathematics

Describe the groups of homomorphisms of the given abelian groups

In algebra we have the following problem to solve: Describe the groups of homomorphisms of abelian groups. (a) $\textrm{Hom}(\mathbb{Q} / \mathbb{Z}, \mathbb{Q})$ (b)

mathematics

Bernoulli's inequality and an unexpected limit

This question is inspired by What would happen to Bernoulli&#39;s inequality if $x&lt;-1$?. Let $x_n=\min\{x\in{\bf R}:(1+x)^n\geq 1+nx\}$, where $n$ is natural and odd (my mistake in the fir

mathematics

Characterizing group homomorphisms

I'm beginning to study group theory and I need some help with this question: Let $f:S_3 \rightarrow \mathbb{C^x}$ be a group homomorphism, where $\mathbb{C^x}=\mathbb{C} \setminus \{0\}$ den

mathematics

mathematics

Botany related Genetically improved Leucaena leucocephala seeds

I am looking for a fast growing Leucaena leucocephala seeds. Now, I am cultivating Leucaena leucocephala which can grow 6.00 Meters in a year. But I am looking for a genetically improved seeds whic...

biology

Words or digits? What is good style for numbers in mathematical writing?

What is considered good style for writing small numbers as words or digits in mathematical texts? I have three concrete examples, are there any differences between those? "M is a matroid of ra

english-language

Some actor's lines from film 'Insomnia'

You do everything around here? Yes. I'm going to cut the deal. Why isn't the phrase just "you do everything here"? I know cut and deal, but it doesn't seem not mean

english-language