# Technical Document Extraction
## Network Diagram (Left Side)
### Nodes
- **Top Layer Nodes**:
- `X_{2,1}` (central node)
- **Intermediate Layer Nodes**:
- `X_{1,1}`, `X_{1,2}`, `X_{1,3}`, `X_{1,4}`, `X_{1,5}`
- **Bottom Layer Nodes**:
- `X_{0,1}`, `X_{0,2}`
### Connections
- **Top-to-Intermediate Connections**:
- `x̃_{1,1,1}`, `x̃_{1,1,2}`, `x̃_{1,1,3}`, `x̃_{1,1,4}`, `x̃_{1,1,5}`
- `x̃_{1,2,1}`, `x̃_{1,2,2}`, `x̃_{1,2,3}`, `x̃_{1,2,4}`, `x̃_{1,2,5}`
- `x̃_{1,3,1}`, `x̃_{1,3,2}`, `x̃_{1,3,3}`, `x̃_{1,3,4}`, `x̃_{1,3,5}`
- `x̃_{1,4,1}`, `x̃_{1,4,2}`, `x̃_{1,4,3}`, `x̃_{1,4,4}`, `x̃_{1,4,5}`
- `x̃_{1,5,1}`, `x̃_{1,5,2}`
- **Intermediate-to-Bottom Connections**:
- `x̃_{0,1,1}`, `x̃_{0,1,2}`, `x̃_{0,1,3}`, `x̃_{0,1,4}`, `x̃_{0,1,5}`
- `x̃_{0,2,1}`, `x̃_{0,2,2}`, `x̃_{0,2,3}`, `x̃_{0,2,4}`, `x̃_{0,2,5}`
### Function Blocks
- **Top Layer Functions**:
- `φ_{1,1,1}`, `φ_{1,1,2}`, `φ_{1,1,3}`, `φ_{1,1,4}`, `φ_{1,1,5}`
- `φ_{1,2,1}`, `φ_{1,2,2}`, `φ_{1,2,3}`, `φ_{1,2,4}`, `φ_{1,2,5}`
- `φ_{1,3,1}`, `φ_{1,3,2}`, `φ_{1,3,3}`, `φ_{1,3,4}`, `φ_{1,3,5}`
- `φ_{1,4,1}`, `φ_{1,4,2}`, `φ_{1,4,3}`, `φ_{1,4,4}`, `φ_{1,4,5}`
- `φ_{1,5,1}`, `φ_{1,5,2}`
- **Bottom Layer Functions**:
- `φ_{0,1,1}`, `φ_{0,1,2}`, `φ_{0,1,3}`, `φ_{0,1,4}`, `φ_{0,1,5}`
- `φ_{0,2,1}`, `φ_{0,2,2}`, `φ_{0,2,3}`, `φ_{0,2,4}`, `φ_{0,2,5}`
## Graph and Equations (Right Side)
### Main Graph
- **Function**: `φ(x)`
- **Equation**:
- `φ(x) = ∑_{i=0}^7 c_i B_i(x)`
- **Time Axis**:
- Range: `t_{-3}` to `t_8`
- Key Points: `t_{-3}`, `t_{-2}`, `t_{-1}`, `t_0`, `t_1`, `t_2`, `t_3`, `t_4`, `t_5`, `t_6`, `t_7`, `t_8`
- **Grid Extension**:
- **Extended Function**: `φ(x) = ∑_{i=0}^{12} c'_i B'_i(x)`
- **Extended Time Axis**:
- Range: `t'_{-3}` to `t'_{13}`
- Key Points: `t'_{-3}`, `t'_{-2}`, `t'_{-1}`, `t'_0`, `t'_1`, `t'_2`, `t'_3`, `t'_4`, `t'_5`, `t'_6`, `t'_7`, `t'_8`, `t'_9`, `t'_{10}`, `t'_{11}`, `t'_{12}`, `t'_{13}`
- **Basis Functions**:
- `B_0(x)`, `B_1(x)`, `B_2(x)`, `B_3(x)`, `B_4(x)`, `B_5(x)`, `B_6(x)`, `B_7(x)`
- Extended Basis Functions: `B'_0(x)`, `B'_1(x)`, ..., `B'_{12}(x)`
- **Coefficients**:
- Original: `c_0`, `c_1`, ..., `c_7`
- Extended: `c'_0`, `c'_1`, ..., `c'_{12}`
### Legend
- **Colors**:
- `B_0(x)`: Red
- `B_1(x)`: Green
- `B_2(x)`: Blue
- `B_3(x)`: Purple
- `B_4(x)`: Cyan
- `B_5(x)`: Orange
- `B_6(x)`: Pink
- `B_7(x)`: Dark Blue
- Extended Basis Functions: Not explicitly labeled but implied by continuation.
### Additional Notes
- **Grid Extension Indicator**: Dashed arrow pointing from original to extended time axis.
- **G Values**:
- `G_1 = 5` (original grid size)
- `G_2 = 10` (extended grid size)
- **Key**: `k = 3` (possibly a parameter for the network or grid configuration).