git clone https://github.com/your-repository/BEC-Net.git
cd BEC-Net


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Equations
This project uses several key equations for simulating the BEC dynamics and encoding the input signal.

1. BEC Dynamics
The time evolution of the BEC is governed by the Gross-Pitaevskii equation, which describes the behavior of the condensate wave function:

iℏ ∂ψ(r,t) / ∂t = (-ℏ² / 2m) ∇²ψ(r,t) + V(r)ψ(r,t) + g |ψ(r,t)|²ψ(r,t)

Where:

ψ(r,t) is the wave function of the BEC.
∇² is the Laplacian operator (spatial derivatives).
V(r) is the external potential.
g is the interaction strength.
|ψ(r,t)|² is the particle density.
2. Input Encoding
The input signal is encoded onto a 2D grid by tiling the input data to fill the grid. The encoding function uses spatial coordinates X and Y:

encoded_input(X,Y) = scale × input_signal_reshaped × (X² + Y²)

Where:

X, Y are the 2D coordinates in the grid.
input_signal_reshaped is the reshaped input signal.
3. Readout Layer
The readout layer uses a linear regression model that maps the extracted features to the target signal:

y(t) = wᵀ x(t) + b

Where:

y(t) is the predicted output.
w is the weight vector.
x(t) are the extracted features at time t.
b is the bias term.



Usage
Prepare Input Data
Generate a sample sine wave signal:

python
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import numpy as np

# Generate a sample sine wave as the input signal
input_signal = np.sin(np.linspace(0, 10 * np.pi, 500))
Run the Simulation
After installing the dependencies, run the simulation with the main.py script:

bash
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python main.py
Example Plots
1. Input Signal
Plotting the original sine wave input signal:

python
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import matplotlib.pyplot as plt

# Plot input signal
plt.plot(input_signal)
plt.title("Original Input Signal")
plt.xlabel("Time")
plt.ylabel("Amplitude")
plt.show()
2. Encoded Input Signal
Visualizing the encoded input signal as a heatmap:

python
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encoded_input = np.random.rand(128, 128)  # Placeholder data
plt.imshow(encoded_input, cmap='viridis', origin='lower')
plt.title("Encoded Input Signal")
plt.colorbar(label="Amplitude")
plt.show()
License
This project is licensed under the MIT License. See the LICENSE file for details.

markdown
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### Explanation of Key Sections:
1. **Equations**: The equations are formatted using LaTeX for clarity. These will be rendered properly in Markdown viewers (like GitHub).
2. **Installation Instructions**: The steps are provided for cloning the repository, installing dependencies, and running the project.
3. **Usage Instructions**: A guide to prepare the input data and run the simulation. Example Python code is provided for clarity.
4. **Example Plots**: Shows how to visualize the input and encoded signal using `matplotlib`.