Refine section titles: remove enumerations and enhance clarity across multiple PDE modules

Author: Daniel Precioso, PhD

.github/copilot-instructions.md

@@ -26,6 +26,7 @@ FOLLOW these rules:
 * Speak directly to the reader using "you" and "your."
 * The output should read clean, concise, and natural—like something a human wrote. 
 * Before finalizing, review and ensure there are no em dashes.
+* Never use enumerations in section titles or headings. Use descriptive headings without prefixes like "Step 1", "1)", or "Track 2".
 
 DO NOT:
 * Use em dashes (only commas, periods, or semicolons are allowed).

modules/networks/index.qmd

@@ -83,7 +83,7 @@ By the end of this session, you should be able to:
 
 ## Flow of this Session
 
-### 1) Fundamentals: graphs in `networkx`
+### Fundamentals: graphs in `networkx`
 
 NetworkX is a Python package for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks [@hagberg2008networkx].
 
@@ -104,7 +104,7 @@ You will practice:
 
 These concepts follow the formal definitions of nodes, links, directed and weighted graphs introduced in the lecture notes.
 
-### 2) Connectivity + centrality
+### Connectivity + centrality
 
 Next we measure structure:
 
@@ -113,7 +113,7 @@ Next we measure structure:
 * clustering coefficient,
 * centrality measures (degree/closeness/betweenness/PageRank) and what they mean.
 
-### 3) Graph models (synthetic networks)
+### Graph models (synthetic networks)
 
 We generate synthetic networks to compare structure:
 
@@ -130,7 +130,7 @@ You will measure:
 * Clustering coefficient.
 * Degree distribution.
 
-### 4) Spreading: SIS and SIR
+### Spreading: SIS and SIR
 
 This is the core computational part of the lab.
 

modules/pde/gierer-meinhardt-1d.qmd

@@ -57,7 +57,7 @@ plt.show()
 plt.close()
 ```
 
-## Step 1: Discretize the Laplacian
+## Discretize the Laplacian
 
 In one dimension, the Laplacian is the second derivative:
 
@@ -94,7 +94,7 @@ def laplacian_1d(values, dx):
 ```
 :::
 
-## Step 2: Choose Boundary Conditions
+## Choose Boundary Conditions
 
 We need one more ingredient before we time step the PDE: what happens at the edges of the domain?
 
@@ -118,7 +118,7 @@ You prescribe fixed values at the endpoints.
 
 In this page we will use **Neumann** boundary conditions because they are easy to interpret in a chemical setting.
 
-## Step 3: Initialize the Fields
+## Initialize the Fields
 
 Start from the homogeneous state and add a small perturbation.
 
@@ -149,7 +149,7 @@ uv += uv * np.random.randn(2, num_points) / 100
 
 Why do we add noise? A perfectly uniform initial condition can hide an instability. Small perturbations reveal whether the system amplifies spatial differences.
 
-## Step 4: Build the PDE Right-Hand Side
+## Build the PDE Right-Hand Side
 
 Use your Laplacian helper for both $u$ and $v$, then add the reaction terms.
 
@@ -186,7 +186,7 @@ def gierer_meinhardt_pde(t, uv, gamma=1, a=0.40, b=1.00, d=20, dx=0.5):
 ```
 :::
 
-## Step 5: Advance in Time
+## Advance in Time
 
 We will use explicit Euler:
 
@@ -222,7 +222,7 @@ for _ in range(num_iter):
 ```
 :::
 
-## Step 6: Plot the Pattern
+## Plot the Pattern
 
 ```python
 import matplotlib.pyplot as plt

modules/pde/gierer-meinhardt-2d.qmd

@@ -60,7 +60,7 @@ plt.show()
 plt.close()
 ```
 
-## Step 1: Discretize the 2D Laplacian
+## Discretize the 2D Laplacian
 
 This is the real jump from the previous page. In 1D, every point had two nearest neighbors. In 2D, every point in the interior has four cardinal neighbors, so the Laplacian becomes a stencil on a grid.
 
@@ -105,7 +105,7 @@ def laplacian_2d(field, dx):
 This stencil assumes periodic wrapping through `np.roll()`. If you want Neumann boundaries, you still need to overwrite the edges after every Euler step.
 :::
 
-## Step 2: Initialize the Grid
+## Initialize the Grid
 
 ```python
 import numpy as np
@@ -134,7 +134,7 @@ uv += uv * np.random.randn(2, num_x, num_y) / 100
 ```
 :::
 
-## Step 3: Build the PDE Right-Hand Side
+## Build the PDE Right-Hand Side
 
 ```python
 def gierer_meinhardt_pde_2d(t, uv, gamma=1, a=0.40, b=1.00, d=20, dx=1):
@@ -169,7 +169,7 @@ def gierer_meinhardt_pde_2d(t, uv, gamma=1, a=0.40, b=1.00, d=20, dx=1):
 ```
 :::
 
-## Step 4: Time Step with Euler's Method
+## Time Step with Euler's Method
 
 Before you animate anything, validate the numerical update with a long loop and one static snapshot. That is the same workflow we used in earlier sessions: get the update rule right first, then wrap it in animation logic.
 
@@ -201,7 +201,7 @@ for _ in range(num_iter):
 ```
 :::
 
-## Step 5: Plot a Snapshot
+## Plot a Snapshot
 
 ```python
 import matplotlib.pyplot as plt
@@ -219,7 +219,7 @@ plt.colorbar(im, ax=ax, label="v(x, y)")
 plt.show()
 ```
 
-## Step 6: Animate the Simulation
+## Animate the Simulation
 
 ```python
 import matplotlib.animation as animation

modules/pde/gray-scott-art.qmd

@@ -8,7 +8,7 @@ Start from the animation on the previous page. Now we will turn it into a small
 
 This is the same pattern you already saw in the interactive pages from earlier sessions: keep the model update fixed, then wrap it with event handlers and controls.
 
-## Step 1: Draw on the Field
+## Draw on the Field
 
 Left click should add a local perturbation and right click should reset a small patch to the base state.
 
@@ -53,7 +53,7 @@ def update_uv(event: MouseEvent, r: int = 3):
 ```
 :::
 
-## Step 2: Allow Continuous Drawing
+## Allow Continuous Drawing
 
 Use three event handlers so the perturbation follows the mouse while the button is pressed.
 
@@ -97,7 +97,7 @@ def on_motion(event: MouseEvent):
 ```
 :::
 
-## Step 3: Pause and Resume with the Keyboard
+## Pause and Resume with the Keyboard
 
 ```python
 from matplotlib.backend_bases import KeyEvent
@@ -121,7 +121,7 @@ def on_keypress(event: KeyEvent):
 ```
 :::
 
-## Step 4: Add Sliders for the Parameters
+## Add Sliders for the Parameters
 
 ```python
 from matplotlib.widgets import Slider
@@ -160,7 +160,7 @@ def update_sliders(_):
 ```
 :::
 
-## Step 5: Put It Together
+## Put It Together
 
 Inside your animation callback, respect the `pause` flag before advancing the PDE.
 

modules/pde/gray-scott.qmd

@@ -51,7 +51,7 @@ plt.show()
 plt.close()
 ```
 
-## Step 1: Start from the Homogeneous State
+## Start from the Homogeneous State
 
 Gray-Scott is usually initialized near the steady state $(u, v) = (1, 0)$, then a small patch is perturbed.
 
@@ -84,7 +84,7 @@ uv[:, c - r : c + r, c - r : c + r] *= 1 + noise
 ```
 :::
 
-## Step 2: Reuse the 2D Laplacian
+## Reuse the 2D Laplacian
 
 We will start with the 5-point stencil. If you implemented the previous page cleanly, you can reuse that helper almost unchanged.
 
@@ -110,7 +110,7 @@ def laplacian(field):
 ```
 :::
 
-## Step 3: Implement the Gray-Scott PDE
+## Implement the Gray-Scott PDE
 
 ```python
 import numpy as np
@@ -145,7 +145,7 @@ def gray_scott_pde(t, uv, d1=0.1, d2=0.05, f=0.040, k=0.060):
 ```
 :::
 
-## Step 4: Time Step and Animate
+## Time Step and Animate
 
 Because `np.roll()` wraps around the array, this implementation already uses **periodic boundary conditions**.
 

modules/pde/turing-instability.qmd

@@ -34,7 +34,7 @@ plt.show()
 plt.close()
 ```
 
-## Step 1: Write the Jacobian at the Fixed Point
+## Write the Jacobian at the Fixed Point
 
 Let
 
@@ -67,7 +67,7 @@ def gierer_meinhardt_jacobian(a=0.40, b=1.00):
 ```
 :::
 
-## Step 2: Implement the Turing Test
+## Implement the Turing Test
 
 A common set of conditions for Turing instability is
 
@@ -113,7 +113,7 @@ If your implementation is vectorized, it should work with scalars and with array
 
 Validate these functions carefully. You will reuse them when you move from 1D to 2D, so this is the theory page that supports the rest of the session.
 
-## Step 3: Plot the Turing Space
+## Plot the Turing Space
 
 Complete the code below to recreate the diagram.
 
@@ -136,7 +136,7 @@ plt.show()
 
 Mark the point $(a, d) = (0.40, 20)$ on your plot. Is it inside the instability region?
 
-## Step 4: Scan Spatial Modes in 1D
+## Scan Spatial Modes in 1D
 
 The Turing conditions say that some spatial mode should grow, but they do not tell you which one. For Neumann boundary conditions on $[0, L]$, the admissible wavenumbers are
 
@@ -187,7 +187,7 @@ def find_unstable_modes_1d(a=0.40, b=1.00, d=30.0, length=40.0, num_modes=10):
 ```
 :::
 
-## Step 5: Extend the Scan to 2D
+## Extend the Scan to 2D
 
 In two dimensions, the modes are pairs $(n_x, n_y)$. On a rectangle with side lengths $L_x$ and $L_y$,