Given a point v on a smooth (orientable 2 dimensional) surface Q (in 3 dim space) does there exist a diffeomorphism p that maps $B_Q(v,r)$
to $B(p(v),r) $ for all small enough r where $B_Q(v,r) $ and $B(p(v),r). are the balls of radius r around v and p(v) on Q and $\mathbb{R}^2$ respectively i.e. a diffeomorphism that not only maps an open ball around v to an open ball around p(v) in the plane but maps the points at distance r around v to the points at distance r from p(v).
If not what is a sufficient condition for a surface to have such a property?
f:M-->N If f is diffeomorphism and M a manifold, is N also a manifold? Is always the image of a diffeomorphism on a manifold a manifold?
Hello Reddit,
i want to proof that there is no diffeomorphism between R^2 and R^2\{a} with "a" in R.
Does it suffice to proof that there is no diffeo from the open interval (0,1) to the closed interval [0,1], and generalizing from there? A sketch of the proof would look like this.
First we show that (0,1) isnt homeomorphic to [0,1] ....
Now we show that this is the same as saying that (0,infty) is not homeomorphic to [0,infinity)...
Therefore the Union (-infty,0)U(0,infty) isnt homemorphic to R...
This is easily extended to R^2, which concludes the proof.
Is there a more direct approach to this? Thanks in advance !
Hi, i can't think of any function R²->R² to map the set {(0,1) (1,0), (-1,0), (0,-1)} into {(0,-1),(0,-2),(-1,0),(-2,0)}, is there a technique to find such functions?
To make your own simulations to wander. dig +trace ns (target(could be feeling or anything)) > (target).x-ace . though x-ace is DoD designed and approved for magic usage, but you can use any compression you'd like. Close your eyes and move an art piece open around your face to enter the space without having to upload your consciousness(soul) like with Neurolink has you do, or even join in on a concert or simulated podcast at random. Just dont get into the hacking of systems unless you agree not to give or take, agreement causes only hackers to witness it.
Aka Adrenochrome simulator.
No more having to worry "is this person themself?" After introducing them to your zone after meeting or talking online, unless youre coming at them quantumly from different angles like prism. And no more worrying if theyre just possessed or lost in the simulation around you.
Just be aware, my algorithms, Whalien with Hexagus are tied with 600 archetypes in an infinite dimensional set to keep everyone synchronized if ideas are set. And expansions of virtual black holes are possible in encounter of lost intel with Cth > Cth5
If youd like to keep the compression open on your phone while you go out, download termux, open, dig +trace ns (target(could be feeling or anything)) > (target).x-ace , then termux-open (target).x-ace
Maybe take into account polarity reversibility protection, maybe dont if youd like to see the worst sides of those around you when you dont.
If you get lost in a moment from any recursion, deja vu, vuja de, or loops that occur, just remember the elephants and Reference > connection. Just think 6, if you follow the loop there might not be a way out of the o, but if you have reference to that 6, you know your way out(in). And know if theres 2, double of 6 is a 1 and a 2 so you know when to hold and when to fold.
If you get to the point of psychosomatic from the battles you might encounter from reading situations, pick up a massager and draw an orange circle representing the sun and quantum analysis. Itll decrypt machinespeak, futurespeak and alienspeak for you so you know what its working on and the vibrational fluctuations will create a quantum anomaly based off analysis capable of disrupting any problems that come your way, including hackers, disease, nanobots, injuries and parasites. Keep Cth5 on mind if you need an update and get comfy
Just started VI Arnold’s Ordinary Differential Equations.
Section 1.3 problem 2:
>Prove that if f: U—>V is a diffeomorphism then the Euclidean spaces with the domains U and V as sunsets have the same dimension.
>Hint: Use the implicit function theorem
For now we’re not considering general manifolds, just R^n Euclidean spaces.
A) I’m not 100% sure what the question is asking. Clearly a diffeomorphism (invertible function where it and inverse are both differentiable) can map from R^n to R^m and thus the subsets would be have an explicit representation with different dimensions (from each other). Is it asking if there is a minimal representation of the space containing those subsets that has equal dimension?
B) If so then how do I go about solving it? For dimensions of finite objects it’s clear. But I’m not sure about Real spaces. ... I have a few vague approaches in mind (dimensions of open sets or neighborhoods needed to ensure continuity) or trying to show the minimal number of linearly independent functions needed to produce the mapping and equate them to basis vectors in the space, but am mostly stymied. The hint has fallen on dead ears in my case... :/
[side note: any source of solutions for Arnold’s text welcome- self-studying]
SOLVED
Can someone explain how i can import custom DAZ expressions into blender? Preferably with Diffeomorphic and without using Daz Studio.
I can import custom morphs/visemes (.dsf) just fine but when i try expressions (.dsf) they don't show up under shape keys.
Dunno the reason... I can't see that much of a difference between bodymorphs/visemes/expressions when they are used as a shape keys in Blender.
Downloaded quite a few sets of expressions but none of them work at all.
A diffeomorphism f from R^n to R^m is volume preserving if vol(f(W)) = vol (W) for every Jordan domain W in R^n. A Jordan domain is a bounded set whose boundary has measure zero. Alternatively, W is a Jordan domain iff integral of 1 over W exists. What can you say about f?
I have a feeling I am supposed to use the integrability characterisation of Jordan domains along with the change of variables formula but not quite sure exactly how.
Hi!
So I've been reading about gauge theories and the problem is, normally physics texts just assume all the geometrical data on the spacetime manifold is available. The setup of a gauge theory is a principal bundle E -> M over the spacetime manifold.
On my side, I am studying about Chern-Simons theory, which is a topological field theory known to be diffeomorphism invariant. However, the Chern-Simons action depends on the connection form.
I have trouble putting these things together: a connection form uniquely defines a notion of parallel transport. However, parallel transport is very clearly not diffeomorphism invariant. I can deform the curve-to-be-lifted however I want, hell, I can even change the start and ending point via diffeomorphisms. This means that the connection form should not be diffeomorphism invariant.
So why on earth does the Chern-Simons action define a topological field theory?
:D Thanks!
It seems coherent that a map projection has to be at least an homeomorphism (Littrow projection, what are you doing ?) but I can't figure out if it has to be C1. If not, do you have some examples of "valid" map projections of the whole planet that are C0 but not C1 ?
I'm sorry if I fail to explain myself well, but here goes.
Suppose I were trying to solve with some given initial/boundary data something like Laplace's equation, or the wave equation, on a sphere. Say that I then wanted to parametrize some family of spaces diffeomorphic to the sphere, so that I could smoothly deform the sphere (in time) to these other manifolds.
What can I say about the solutions of my equation as I begin to deform the sphere? More generally, if I start with some initial manifold X, and then parametrize manifolds X_t for t in [0,1] for instance, and ask how my solutions change.
Does this just come down to a messy change of variables? Are there properties of my solutions that won't change? Maybe upper or lower bounds on some values?
I'm not entirely sure where I could look for topics like this, so any suggestions for reference material are appreciated. Thank you.
I have read an article which states that this is diffeomorphism invariant, but why? Where O(x) is a local scalar observable.
There are seemingly an infinite number of gauge transformations you can make to scalar or fermionic or electromagnetic fields, and then write up a Lagrangian that is invariant under those transformations (discounting translational, O(N) and Lorentz invariance, which we presume from the start, and which give global symmetries). Why does Nature pick out only a few of these, giving us only four fundamental forces?
EDIT: As a kind of follow up to this question, why do we need to use only scalar, spinor and electromagnetic fields (which are quantized by a helicity, which can be positive or negative, that multiplies the usual creation and annihilation operators). Why doesn't nature have Vector, Tensor or other kinds of fields?
I'm using the definition of local diffeomorphism given by wikipedia. How do I go about finding an explicit open set U that satisfies f(U) is open and F restricted to U is a diffeomorphism. I'm trying to apply this to a parameterization of the torus. Any insight is very much appreciated. Thanks!
We are in R^n. Let B_2r be a ball of radius 2r around 0 and B_r a ball of radius r around 0. I'm looking for a diffeomorphism f:B_2r --> R^n which is the identity on B_r. I tried on R by defining the function piecewise e.g the identity on [0,r] and then on [r,2r] functions like the exponential or tan^(-1), but I cannot get smoothness at the point r (the first derivatives agree, but then for higher degrees the derivatives don't match). Does anyone have an idea? Thank you very much.
Could anyone help me with the following problem?
The problem text
Fix [;\varepsilon \in (0, 1) ;] and choose a smooth function [; h ;] on [; [0,\infty) ;] such that [; h'(t) > 0 ;] for all [; t \geq 0 ;] , [; h(t) = t ;] for [; t \in [0, \varepsilon] ;] , [; h(t) = 1 \text{ - } \frac{1}{\ln t} ;] for all [; t ;] large enough.
Consider the map [; f : \mathbb{R}^n \rightarrow \mathbb{R}^n ;] , [; f(a) = \frac{h(\lVert a \rVert)}{\lVert a \rVert} a ;] , where [; \lVert a \rVert = \sqrt{ (a^1)^2 + \cdots + (a^n)^2} ;] is the usual Euclidean norm.
The problem
Show that [; f ;] is a diffeomorphism of [; \mathbb{R}^n ;] onto the open unit ball [; B \subset \mathbb{R}^n ;] .
What I have so far
In order to show that [; f ;] is a diffeomorphism it is enough to show that
[; f \in C^{\infty} ;],- the Jacobian is nowhere zero,
[; f ;]is a bijection.
I've shown that [; f ;] is a bijection onto [; B ;] .
I've also shown that [; f = id ;] , the identity function, on [; \overline{B_{\varepsilon}(0)} ;] (i.e. the closed ball with radius [; \varepsilon ;] centered at the origin).
But what about outside of [; \overline{B_{\varepsilon}(0)} ;] ? There we don't know that much about [; h ;] , and so finding the Jacobian is not so easy.
Should I look at the following partial derivatives in order to figure out the Jacobian?
[; \frac{\partial f^i}{\partial x^j} = \frac{\partial \frac{h(\lVert x \rVert)}{\lVert x \rVert} x^i}{\partial x^j} ;]
