All Questions
11 questions
7
votes
1
answer
271
views
Closest point in $SU(n) \otimes SU(n)$ to $SU(n^2)$
What is the closest $V_1 \otimes V_2 \in SU(n)\otimes SU(n)$ in the squared trace inner product to a given $U \in SU(n^2)$? I.e. minimize over $V_1, V_2$:
$\min_{V_1, V_2} | V_1 \otimes V_2 - U|$ in ...
- 2,099
3
votes
1
answer
115
views
Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?
Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
- 2,099
3
votes
1
answer
134
views
Singular curves of affine distributions on a Lie group
Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?
Specifically the case of a right invariant affine distribution: $D_{U} = \{...
- 2,099
3
votes
0
answers
70
views
Attainability of Global Optima In Optimal Control
Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...
- 2,099
2
votes
1
answer
236
views
Simultaneous integral equation on $SU(n)$
Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves:
$\frac{d U_s}{ds} = (a + w(s)b)U_s$
for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
- 2,099
2
votes
0
answers
94
views
Maximize a tricky function on $SU(n)$
Given non-zero $\xi \in \mathfrak{su}(n)$ and $G \in SU(n)$, consider the function:
$Q(U) = Tr(G^{\dagger}U)GU^{\dagger} - Tr(U^{\dagger}G) UG^{\dagger}$
(which just happens to be the gradient of $|...
2
votes
0
answers
115
views
Are singular critical points isolated for control systems on compact semisimple Lie groups
Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...
- 2,099
1
vote
1
answer
172
views
Existence of a Lie algebra element orthogonal to the adjoint orbit of another element
Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...
- 2,099
0
votes
1
answer
138
views
Gradient on $SU(n)$
I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
- 2,099
0
votes
1
answer
130
views
equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
- 2,099
0
votes
0
answers
71
views
Curves in $\mathfrak{su}(n)$ with specific property
Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...
- 2,099