Quantum Hamiltonian reduction and Quantum Airy structure

I have been reading Kontsevich and Soibelman’s “Airy structures and symplectic geometry of topological recursion” (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section 7.2 on Quantum Hamiltonian reduction. In particular, I’d like to understand how to compute $ \psi_{\hat{\mathcal{B}}}$ .

The following is what I think the paper says: Let $ (W, \omega)$ be a symplectic vector space, $ G\subset W$ a coisotropic subspace ($ G^\perp \subset G$ ) and $ L \subset W$ a Lagrangian submanifold. Given a point $ x \in L \cap G$ such that $ T_xL + T_xG = T_xW$ then $ \mathcal{H} := G/G^\perp$ is a symplectic vector space and $ \mathcal{B}_x := L_x \cap G \hookrightarrow \mathcal{H}$ is embedded as a Lagrangian submanifold (I interpreted the germ $ L_x$ as ‘small neighbourhood of $ L$ around $ x$ ‘, which is probably wrong?). Then $ G$ is naturally embedded into $ \mathcal{H}\times \bar{W}$ , where $ (\bar{W},\omega) = (W,-\omega)$ as a Lagrangian subspace. Let the coordinates of $ W$ be $ (q,p)$ and the coordinates of $ \mathcal{H}$ be $ (q’,p’)$ . From general theory (Section 2.4?) we have the wave function $ \psi_{G}(q,q’) = \exp(Q_2(q,q’)/\hbar)$ quantizing $ G$ where $ Q_2$ is a quadratic polynomial. Then the Hamiltonian reduction of $ L\subset M$ to $ \hat{\mathcal{B}}_x \subset \mathcal{H}$ at the level of wave functions becomes \begin{equation} \psi_{\hat{\mathcal{B}}}(q’) := \int \psi_{G}(q,q’)\psi_L(q) \end{equation} where $ \psi_L(q)$ is the wave function quantizing the quadratic Lagrangian $ L$ as studied in Section 2.4, 2.5.

My Attempts

  1. The only natural way I can think of to embed $ G \hookrightarrow \mathcal{H}\times \bar{W}$ is by writing $ G = T_x\mathcal{B}\oplus V$ , where $ V$ is a Lagrangian complement to $ T_xL$ , then embed $ G$ via $ T_x\mathcal{B}\hookrightarrow \mathcal{H}, V \hookrightarrow \bar{W}$ .
    However, this would mean I can write $ Q_2(q,q’) = Q_{T_x\mathcal{B}}(q’) + Q_W(q)$ . Evaluating the integral we would get $ \psi_{\hat{\mathcal{B}}} = \text{constant}\times \exp(Q_{T_x\mathcal{B}}(q’))$ . So $ \psi_{\hat{\mathcal{B}}}$ only going to quantize $ T_x\mathcal{B}$ in $ \mathcal{H}$ for a choice of $ x$ instead of the entire Lagrangian submanifold $ \mathcal{B}\subset \mathcal{H}$ as I would expect the result of this section to be about.

  2. Perhape the embedding $ G \hookrightarrow \mathcal{H}\times \bar{W}$ meant to be such that the image in $ \mathcal{H}$ is actually $ \mathcal{B}_x$ (and the image in $ \bar{W}$ is $ V$ ). If that is the case then $ G$ is embedded as Lagrangian submanifold not subspace (as stated in the paper). But then I’m still going to have $ Q_2(q,q’) = Q_{\mathcal{B}}(q’) + Q_W(q)$ where $ Q_{\mathcal{B}}(q’)$ is no longer just quadratic in $ q’$ and probably can be found using Section 2.4. But then I’m still going to have $ \psi_{\hat{\mathcal{B}}} = \text{constant}\times \exp(Q_{\mathcal{B}}(q’))$ which make me wonder why don’t I just directly quantizing $ \mathcal{B}_x \subset \mathcal{H}$ since the start instead of looking at $ G\hookrightarrow \mathcal{H}\times \bar{W}$ and do a quantum Hamiltonian reduction. Quantizing $ \mathcal{B}_x \subset \mathcal{H}$ directly seems difficult and I thought Hamiltonian reduction will help me with it.

Obviously, I have missed a lot of important things. If someone could help me understanding this section better or guid me to good references for quantum Hamiltonian reduction I would be really appreciated. Thank you.

Why my python code fails due to time limit exceeded? (Kattis prime reduction challenge)

I am solving the prime reduction challenge on Kattis (https://open.kattis.com/problems/primereduction). I’m already solve it using Java. Now I am trying to solve it on python. My code did well on the sample tests, but on the second test my code fails because time limit exceeded.

Someone could tell me what I did wrong?

My python code:

    import sys from math import sqrt, floor  def is_prime_number(number):     if number == 2:         return True     if number % 2 == 0:         return False     return not any(number % divisor == 0 for divisor in range(3, floor(sqrt(number)) + 1, 2))  def sum_prime_factors(number):     factors = 0     while number % 2 == 0:         factors += 2         number = floor(number / 2)     if is_prime_number(number):         factors += number         return factors     for factor in range(3, floor(sqrt(number)) + 1, 2):         if is_prime_number(factor):             while number % factor == 0:                 factors += factor                 number = number / factor             if number == 1:                 return factors      factors += number     return factors  def print_output(x, i):     if is_prime_number(x):         print(x, i)         return     i = i + 1     factors = sum_prime_factors(x)     print_output(factors, i )  [print_output(item, 1) for item in map(int, sys.stdin) if item != 4] 

dimensionality reduction expectaion

I m studing dimensionality reduction (SVD in particular) and i see this question from topic.

Assume we have a vector $ x \in R^d$ and consider $ F(X)=s^t x$ , where $ s$ is a d-dimensional random vector with entries drawn uniformalily independently from $ [-1,1]$ .

What is the value of $ E[F(x)^2] $

I m starting now from zero, so i m need to study more. The exercise ask a formal prof, but i wish just understand the philosophy. I see in every question that choose uniformly and independently from [-1,1]… this is take because take a symmetrical range?the expectation in this range is often $ 1/2$ ?

Adapting a function for amplitude reduction based on distance.

I was wondering how the amplitude of the function (a consequent signal) is affected based on the based distance. For instance, given an electrode, how is the amplitude of a function (the source) affected?

Example of a source and electrode

Thus, how would I adapt the following Gaussian function to account for distance:

Gaussian Function

I would very much appreciate any assistance.

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SQ LITE reduction of size for embedded system

I have been working on this project for quite a while, while SQ Lite is the smallest database system for all uses, it is quite big and bulky for embedded. So I am looking for a way to reduce the size of its library. Is there a way to remove all the join functionalities? As my project only uses SQ Lite as a source of faster file access and storage. With only one database used and only the functions of select, insert, and delete used. Sorry new to using SQ Lite.