Differential Equations. math53.stanford.edu.

Logistics

Prof. Rafe Mazzeo

TAs

• Rodrigo Angelo
• Zhenyuan Zhang

Assignments

• Pre-lecture reading + questionnaire
• PSets: Wed 9A
• 2 Midterms + 1 Final: wk 4 + wk 7, Thurs Evening; Tuesday 12:15

Review

1. it suffices to study First Order ODEs because we can convert all higher order functions into a First Order ODEs
2. homogeneous linear systems \(y’=Ay\) can be solved using eigenvalue, matrix exponentiation, etc. (recall that special cases exists where repeated eigenvalues, etc.)
3. inhomogeneous systems \(y’ = Ay +f(t)\) can be solved using intergrating factor or variation of parameters method
4. general analysis of non-linear \(y’=f(y)\): we can talk about stationary solutions (1. linearize each \(y_0\) stationary solutions to figure local behavior 2. away from stationary solutions, use Lyapunov Functions to discuss), or liapenov functions
5. for variable-coefficient ODEs, we decry sadness and Solving ODEs via power series

Content

• Ordinary Differential Equations
• Partial Differential Equations

What we want to understand:

What if we can use diffusion models to generate Laproscopic surgeries to train surgeons?

Problem

Asking dalle to just “generate a Laproscopic surgery” is not going to work. It will give you cartoons.

Approach

1. text problem formulation: “grasper grasp gallbladder”
2. encode text into latents
3. do diffusion with late fusion of latents

Data: Cholec T-45

Weighting

Scoring: Perception Prioritized Weighting + Prioritization for Signal-to-Noise

(Ho et al, 2020)

Text

“[subject] [verb] [object] [surgical phase]”