Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space.
What I have tried:
If we can find a finite subgroup $G\subset Gal (\mathbb C/\mathbb Q)$ such
that $G$ contains the complex conjug...
In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding original open set.
The following facts are known about shrinking spaces:
Every shrinking space is normal.
Every shrinking space is countably paracompact.
In a normal space, every locally finite, and in fact, every point finite open cover admits a shrinking.
Thus, every normal metacompact...
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that:
1.
∅
∈
I
{\displaystyle \emptyset \in I}
2.if
A
∈
I
...
The Baire category theorem (BCT) is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.
The theorem was proved by René-Louis Baire in his 1899 doctoral thesis.
== Statement of the theorem ==
A Baire space is a topological space with the following property: for each countable collection of open dense sets
{
U
n
}
n
=
1
...
This problem actually has three parts, and I have questions about each part. (What is in block quotes is the actual text of the problem.)
Part (a):
Consider the vector field on the circle given by $\dot{\theta} = \cos
\theta$. (Note: $\dot{\theta} = \frac{d \theta}{dt}$) I need to show
t...
Consider the transforms
$$ A(f(x),g(x)) = h(x) = 1 + \int_0^x f(x - t) g(t) dt $$
$$ B(h(x),g(x)) = f(x) $$
$$ C(h(x),f(x)) = g(x) $$
Where the functions on the LHS are given.
Notice $B,C$ are the inverses of $A$.
How to compute $B,C$ ??
Is there an integral representation for $B$ or $C$ ?
...
