All Questions
Tagged with ca.classical-analysis-and-odes lie-groups
18 questions
-1
votes
0
answers
114
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
1
vote
0
answers
205
views
Fourier transform of functions mapping manifolds, is there a definition?
$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form
$$
f : \mathbb{R} \to \SO(3)^n
$$
Since $\SO(3)$ is a compact group so is $\SO(3)^n$.
Now if ...
- 115
4
votes
1
answer
364
views
When is a smooth field's flow map volume preserving diffeomorphism
Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a (single) real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi_V$ defined as ...
- 43
2
votes
0
answers
122
views
Invariance of $\mathrm{SO}(3)$ dynamics when expressed via Euler angles
$\DeclareMathOperator{\SO}{\operatorname{SO}}$I am trying to understand the properties of the $\SO(3)$ Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions.
I am ...
- 21
1
vote
0
answers
36
views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 5,405
-1
votes
1
answer
181
views
Reparameterization and group structure
I ran into the following question; let $x,y$ be two points in $\mathbb{R}^d$. Let $(\psi_t)_{t\geq 0}$ be the mapping from $\mathbb{R}^{2d}$ to $\mathbb{R}^{2d}$ defined, for all $t\geq 0$, by
$$
\...
user153086
1
vote
0
answers
126
views
What is the unitary $1$-parameter group generated by a vector field on a manifold?
If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
- 5,407
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
4
votes
1
answer
509
views
Lie Symmetries of the Bessel Differential Equation
The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...
- 22.8k
4
votes
1
answer
1k
views
Relationship between Laplacian and Hessian on compact Lie groups
If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$
where $\Delta$ ...
- 195
1
vote
1
answer
121
views
A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]
Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...
- 183
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
4
votes
0
answers
238
views
A possible refinement of a theorem of Malliavin
Due to a theorem of Malliavin (an improvement of an erlier theorem of Dixmier and Malliavin, see here) we know that every compactly supported smooth function $f$ on $\mathbb R^n$ can be written as a ...
- 85.6k
1
vote
0
answers
106
views
Semiflows and continuous symmetries
Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = F(T_\...
- 268
2
votes
3
answers
549
views
Non-continuous representations of $\operatorname{SL}_2(\mathbf{R})$
How does one construct a non-continuous representation $\rho:\operatorname{SL}_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?
- 7,609
11
votes
1
answer
664
views
Lie $2$-groups and differential equations
I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: ...
- 11.8k
23
votes
1
answer
2k
views
Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?
In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is ...
- 53.3k
7
votes
2
answers
1k
views
Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
- 54.6k