Lie algebras of infinite dimensional Lie groups

I have to work with Lie algebras of some infinite dimensional ‘Lie groups’ (e.g. $ \Omega SL_2(\mathbb{C})$ ) but i’m not sure on how to approach infinite dimensional groups, for loop group it is not so obvious what should be considered a neighborhood of the identity. I don’t want to see the whole solution, maybe just some hints, but much more i would appreciate an explanation (or some reference) of how one should view local structure of such groups.

One Dimensional Japanese Puzzle

0

So I have been practicing competitive coding for a while now and am using codeforces.com for the same. I recently encountered a problem called “One Dimensional Japanese Crossward” on that site ( https://codeforces.com/problemset/problem/721/A ). After lots of tries and lots of fails, I was finally able to get the code accepted by the judge. However, for some reason I feel that my code is really inefficient and there might be an easier and better way to solve this problem.

The problem is a such: The judge inputs an integer and then a string with those many characters being either ‘B’ or ‘W’. We need to output how many groups of ‘B’ there are and how many ‘B’s there are in each group. For example:

Input:

6

BBWBWB

Output:

3

2 1 1.

This is what my code looks like:

#include<iostream> #include<vector> using namespace std;  int main() {     int n;     cin >> n;     string s;     cin >> s;      int bGroups = 0;     int wGroups = 0;      char initChar = s[0];      for (int i = 1; i < s.length(); i++)     {         if (initChar == 'B')         {             if (s[i] != initChar)             {                 bGroups++;                 initChar = 'W';             }         }         else if (initChar == 'W')         {             if (s[i] != initChar)             {                 wGroups++;                 initChar = 'B';             }         }      }      if (s[n - 1] == 'B')     {         bGroups++;     }     else if (s[n - 1] == 'W')     {         wGroups++;     }      char init = s[0];     vector<int>grpSize(bGroups);     int counter = 0;     int i = 0;     while (counter < bGroups)     {         if (init == 'B')         {             while (init == 'B')             {                 grpSize[counter]++;                 i++;                 init = s[i];             }             counter++;             while (init == 'W')             {                 i++;                 init = s[i];             }         }         else         {             while (init == 'W')             {                 i++;                 init = s[i];             }             while (init == 'B')             {                 grpSize[counter]++;                 i++;                 init = s[i];             }             counter++;         }     }      cout << bGroups << endl;     for (int i = 0; i < bGroups; i++)     {         cout << grpSize[i] << " ";     } } 

So basically, in the first pass over the string, I count how many groups of ‘B’ and ‘W’ there are. Then I create an int vector to store the size of each ‘B’ group. Then on the second pass I fill in the values into the vector and then output the results.

If you think you have a more efficient way to solve this then please drop it down below. Thanks a lot!

Complete positivity with infinite dimensional auxillary spaces

The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo’s Statistical Structure of Quantum Theory) is that a linear map between (sub $ C^*$ algebras of) the bounded operators on some Hilbert spaces $ \phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})$ is $ k$ positive if the map \begin{align} \left(\mathrm{id}_k\otimes\phi\right):\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{K}), \end{align} you get by tensoring with the identity on $ \mathcal{L}(\mathbb{C}^{k})$ is positive, and $ \phi$ is completely positive if it is $ k$ positive for all $ k\in\mathbb{N}$ .

My general question is why do we consider all finite dimensional auxillary spaces $ \mathbb{C}^k$ , rather than infinite dimensional dimensional spaces? In particular from the point of view of quantum information theory we use complete positivity to ensure we can “act locally” on our system of interest whilst the global state remains positive. It seems natural to want this to happen even if the global state is infinite dimensional. Note that this is only an interesting question if $ \mathcal{H}$ and $ \mathcal{K}$ are infinite dimensional.

For convenience I will call maps $ \phi$ such that \begin{align} \left(\mathrm{id}_{\mathcal{S}}\otimes\phi\right):\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{K}), \end{align} is positive for every Hilbert space $ \mathcal{S}$ , where $ \mathrm{id}_{\mathcal{S}}$ is the identity on $ \mathcal{L}(\mathcal{S})$ , extra completely positive. The question may make more sense if we restrict this to separable $ \mathcal{S}$ .

It is relevant that the Stinespring factorisation is possible if, and only if the map is completely positive in the usual sense.

My specific questions are

  1. Is there already a name for, and work on, the extra completely positive maps in the literature? After some searching, and asking colleagues I have not found anything about them.
  2. Is there a $ \phi$ which is completely positive but not extra completely positive? Conversely I would be very interested in a proof that all completely positive maps are extra completely positive. I have tried to come up with an example of the former, and a proof of the latter but with no success.
  3. If the extra completely positive maps are a proper subset of the completely positive maps is there a nice characterisation of them (e.g. a “Stinespring-esque” factorisation)?

How to compute the amplitude of a sampled wave in three dimensional space

I have a sine or cosine wave in a random direction in 3D space and in a random orientation. I want to calculate the amplitude and axis for such a wave. I am sampling the wave at a high rate, compared to the frequency of the wave. I am getting the 6 DOF values for each sample.

I am looking for pointers on how to compute the amplitude and axis of the wave.

Two dimensional motion with variable acceleration

I’m doing a question in a textbook which is known to sometimes have wrong answers. However, it’s more likely that I’m just being stupid.

Particles P and Q, each of mass 0.5 kg, move on a horizontal plane, with east and north as i and j directions. (x and y).

Initially, P has velocity (2i -5j)m/s and Q is travelling north at 2m/s. Each particle is acted on by a force of magnitude t Newtons.

The force on P acts towards north-east, while that on Q acts towards the south-east. Work out the value of t for which:

a. The two particles have the same speed.

b. The two particles are travelling in the same direction.

First of all I’m confused about t. I’m sure t in the question is the force, and can be split into components of the force. However in the mark scheme I’m told to integrate with respect to t, which implies t means time.

For part a I integrated the acceleration with respect to t anyway and got the velocity at a time. Then I understand that you can find the constants for both velocity of P and Q and then equate them to find the time when they both have the same speed. I managed to get to the answer in the textbook which is 4.2 seconds.

Now part B is where I’m completely stuck and the mark scheme has no explanation of what they are doing to get the answer which is apparently 2.11 seconds.

How do you show that two particles are travelling in the same direction and how can I find the time when this occurs?

Is the criteria such that the velocities need to be in the same direction? Any help would be appreciated.

Higher dimensional analog of a rational elliptic surface: reference request

Consider two triples of lines $ L_0, L_1, L_2$ and $ M_0, M_1, M_2$ in $ \mathbb{P}^2.$ A blow up of $ \mathbb{P}^2$ at 9 points $ L_i \cap M_j$ is a rational elliptic surface, a very well understood object in algebraic geometry.

Here is a natural higher dimensional generalization. Consider $ 2n+2$ hyperplanes $ L_0, \ldots, L_n$ and $ M_0, \ldots, M_n$ in $ \mathbb{P}^{n}$ in general position. A blow up of $ \mathbb{P}^n$ at $ (n+1)^2$ codimension $ 2$ subspaces $ L_i \cap M_j$ is a rational variety, which admits a structure of a Calabi-Yau fibration over $ \mathbb{P}^1.$

Question: What is known about sections and singular fibers of this fibration? What is known about the structure of the Mori cone of this space?

This must be a well understood object, I will be very grateful for any references.

positive definiteness of matrix based on lower dimensional positivity

Let $ A$ be an $ n \times n$ symmetric matrix with real coefficients and $ k \leq n$ . Of course $ A$ is positive on $ \Bbb R^n$ if it is positive definite on all $ k$ -dimensional subspaces of $ \Bbb R^n$ .

I am looking for results that aim to optimize this simple phenomenon as I now try to explain somewhat vaguely.

Something along the lines of: it is enough to check that $ A$ is positive definite along a certain subclass of $ k$ -dimensional subspaces, that is much smaller then the space of all $ k$ -dimensional subspaces. If there are results where you can actually pick a very specific finite set of $ k$ -dimensional subspaces (perhaps also having extra conditions on $ A$ in the statement), that is even better.

The question is intentionally vague because at this point I simply want to learn about the existence of such results.

Can the Dimensional Lock spell keep a creature stuck in a Maze spell?

Suppose a spellcaster casts Maze on a victim. On the next round, while the victim is in the maze, the spellcaster casts Dimensional Lock on the victim’s original location. If the Maze duration runs out or the victim escapes, but the Dimensional Lock is still active, what happens?

The Maze spell says the creature would be freed and returned to its original location:

On escaping or leaving the maze, the subject reappears where it had been when the maze spell was cast.

However, the Dimensional Lock spell specifically prevents travel via Maze:

You create a shimmering emerald barrier that completely blocks extradimensional travel. Forms of movement barred include astral projection, blink, dimension door, ethereal jaunt, etherealness, gate, maze, plane shift, shadow walk, teleport, and similar spell-like abilities. Once dimensional lock is in place, extradimensional travel into or out of the area is not possible.

Does this mean the victim cannot return until Dimensional Lock runs out? If not, and the victim escapes the maze while the Dimensional Lock is active, then where do they go?