Search found 284 matches
- Tue Sep 27, 2016 4:19 pm
- Forum: Complex Analysis
- Topic: Are These Sets Quasi-Conformal?
- Replies: 0
- Views: 2832
Are These Sets Quasi-Conformal?
This post is related to this one: http://mathimatikoi.org/forum/viewtopic.php?f=4&t=88&hilit=biholomorphic In the above post we saw that the following sets are not conformally equivalent: The unit disc $ \mathbb{D} $ and the complex plane $ \mathbb{C} $. The punctured unit disc $ \mathbb{D}^...
- Wed Aug 31, 2016 1:45 pm
- Forum: Algebra
- Topic: An Interesting Exercise
- Replies: 4
- Views: 8478
Re: An Interesting Exercise
1) If X is compact then is not subspace 2)The functions in Hahn-Banach theorem are linear 1) As $X$ is a subset of $ \mathbb{R}^{n} $, it is naturally a subspace; We additionally require it to be compact. 2) Hahn Banach Theorem/ Formulation What can you say about the solution of the exercise? How i...
- Sat Jul 30, 2016 6:31 pm
- Forum: Algebraic Geometry
- Topic: Trivial Line Bundle
- Replies: 0
- Views: 2928
Trivial Line Bundle
Let $X$ be a complete variety and let $ \mathcal{L} $ be a line bundle on $X$. The following are equivalent:
- $ \mathcal{L} $ is trivial.
- $ \mathrm{H}^{0}(X,\mathcal{L}) \neq 0 \neq \mathrm{H}^{0}(X,\mathcal{L}^{*}) $.
- Sat Jul 30, 2016 11:27 am
- Forum: General Topology
- Topic: Semicontinuity
- Replies: 0
- Views: 5705
Semicontinuity
Definition : Let $Y$ be a topological space. A function $ \varphi \ \colon Y \longrightarrow \mathbb{Z} $ is called upper semicontinuous if for every $y \in Y$ there exists an open neighborhood $U$ of $y$ such that $ \varphi(y) \geq \varphi(y^{\prime}) $ for all $ y^{\prime} \in U $. Show that a fu...
- Thu Jul 28, 2016 11:42 pm
- Forum: Algebraic Structures
- Topic: Question about tensor products
- Replies: 0
- Views: 2642
Question about tensor products
Can the tensor product of a free and a non-free module be a free module?
- Thu Jul 28, 2016 4:46 pm
- Forum: Algebraic Structures
- Topic: Not A Direct Sum
- Replies: 0
- Views: 2291
Not A Direct Sum
Show that a local ring cannot be a direct sum of two other rings.
- Thu Jul 28, 2016 4:45 pm
- Forum: Algebraic Geometry
- Topic: On Étale Morphisms
- Replies: 0
- Views: 2878
On Étale Morphisms
Consider the morphisms of schemes $ f \ \colon X \longrightarrow Y $ and $ g \ \colon Y \longrightarrow Z $. If $ g \circ f $ is étale and $g$ is unramified, then show that $f$ is étale.
- Sun Jul 24, 2016 11:34 pm
- Forum: Differential Geometry
- Topic: Algebraic Tangent Space
- Replies: 0
- Views: 4324
Algebraic Tangent Space
Let $M$ be a smooth manifold. For a point $p \in M$, set \[ \mathfrak{m}_{p} = \left\{ \ f \in C^{\infty}(M) \ \big| \ f(p) = 0 \ \right\} \]Show that $\mathfrak{m}_{p}$ is a maximal ideal of $C^{\infty}(M)$. Any derivation on $ C^{\infty}(M) $ is determined by its values of $ \mathfrak{m}_{p} $. Th...
- Sun Jul 24, 2016 11:20 pm
- Forum: Differential Geometry
- Topic: Trivial Line Bundle
- Replies: 0
- Views: 4242
Trivial Line Bundle
Show that the determinant line bundle $ \det \left( T^{*} \mathbb{S}^{n} \right) $ is trivial.
- Fri Jul 22, 2016 7:48 pm
- Forum: Differential Geometry
- Topic: Injective Immersion Vs Embedding
- Replies: 0
- Views: 4170
Injective Immersion Vs Embedding
Show that a closed, injective, continuous map is a (topological) embedding. Give an example to show that an injective immersion can fail to be an embedding. Show that an injective immersion $ F \ \colon M \longrightarrow N $ (between smooth manifolds) is a (smooth) embedding if either $M$ is compac...