Computational Geometry

Thu, 01 Oct 2020 00:00:00 +0000

Simple Notes for Computational Geometry

Numerical Analysis

Mon, 20 Jan 2020 00:00:00 +0000

Table of Contents Sheet / Course Map for Numerical Analysis

Math Modelling

Mon, 20 May 2019 00:00:00 +0000

Important Topics

Routh Hurwitz Test
Matrix of Variation/Jacobian of dx/dt (and dy/dt)
Stability of solution
- Local Stability (Matrix of Variation) Eigenvalues
  - Positive Definate : Unstable
  - Negative Definate : Local AS
  - Imaginary Eigenvalues: Spiral
- Global Stability (Appropriate Lyaponov Function)
  - Test stability by Lyapunov Funcn or any other (V) has derivative negative definate

$$V(x) = x - x^* - x^* ln(\frac{x}{x^*}) \frac{k_1}{2}(T- T^*)^2 + \frac{k_2}{2}(U-U^*)^2$$$$\frac{dV}{dt} = \frac{\dot{x}}{x}( x - x^*) + other$$

Quick Finding of Eigenvalues (see Prerequisites)
Complex Eigenvalues and Calculation of Spiral
Linearization of Solution
Logistic regression Model $$ \frac{dx}{dt} = rx(1 - \frac{x}{k})$$
Persistance / Permanance of Solution
Picard-Landlof Theorem - Existence of Solution
Sylvester’s Criteria - b^2 - 4ac conditions
Hamiltonian $$ H(x, t, u, \lambda) = g*{divident}(x, t, u) + \lambda f*{capital\ assets}(x, t, u) $$
Pontrayagin’s Maximum Principle $$ \frac{d\lambda}{dt} = -\frac{dH}{dx}$$
Bang-Bang and Singular Control (Control Theory)
De Carte’s Rule of Sign
Dulac Bendixson Criteria for periodicity of soln
Bionic Equilibrium Conditions (for Optimal Harvesting)
Hopf Bifurcation: The point where behavior of system stability changes. Opposite stability before and after critical value.
Lebesgue Cycle Stability
Basic Reproduction No.
LimSup Method for showing boundedness
Standard Comparison Theorem, Amax > Bmin, well-posedness …

Models

Malthusian Growth Model $$ dx/dt = rx$$
Logistic Growth Model (inter-specific interference)
Resource-Consumer and similar models -
- Prey-Predator Model (or Resource Consumer)
  - specialized prey-predator
  - generalized prey-predator
- Competetive Model
- Cooperation Model
3 Species Food Chain Model (Logistic Growth with interspecie interface)
Opimal Harvesting (fish) Model - Max Sustainable Yield
Migration of Fishes Model
Pollution Toxicant Models-
- 2D Model
- 3D Model - Uptake of Conc (POST-Midsem)
Susceptible-Infected and variant models
- SI Model
- SIS Model(with immunity)
- SIR Model(with complete cure forever)
- SEIR Model(Both Exposed and Recovery types)

Analysis of Solution

Boundedness
Positivity and Solution Space $\Omega$
Persistance of Solution (Show Lower Bound)
Periodicity or not (Dulac Bendixson Criteria)
Equilibrium Points
Local Stability Analysis
- Linearize solution and then find values OR…
- Use generalized matrix of variation and plug in values
Global Stability Analysis
- Choose Lyapanov Function and terms based on Logistic Growth or not
- Differentiate and show Negative Definate
- Use Sylvester’s Criteria and compare terms using Routh-Hurwitz Criteria
- Routh Array to show stability

Other Analysis Techniques

Rate $\dot{r}$ for growth and $\dot{\theta}$ for clockwise/anticlockwise in spirals
Lebesgue Cycle stability in spirals
Critical points in Hopf Bifurcation and stability chart
Basic Reproduction Number Calculation

Sample Model to check equilibrium points

graph TD A[Formulate the Rate Diffn Equations] --> AB AB[ Find Omega. Check if bounded with limsup and show positive also. Check Persistance/Periodicity] --> B B[Find equilibrium points where rate = 0] --> C C[Local Stability. Get matrix of variation/Jacobian at these points] --> D C --> E[Get Char Eqn with Eigenvalue] E --> F[Use Routh Hurwitz to get roots' sign] F --> G[From sign of eigenvalues determine stability of Local Solution] D[Get Eigenvalues of the matrix of variation] --> G G --> H H[Global Stability. Use Lyapunov function variants Derivative] --> I[Check if negative definate by adding/subtracting]

Harvesting Model Sample Flow

Steps for Solving

Principles of Economics

Mon, 20 May 2019 00:00:00 +0000

Introduction

These are my notes for POE. The notes were written using markdown and vim (see iamcco/markdown.nvim plugin) and Pandoc (with eisvogel template). Feel free to collaborate to the Online Version of POE Notes. These are only supplementary notes and NOT Lecture Notes. Some useful resources.

Graphs and Networks

Sun, 10 Mar 2019 00:00:00 +0000

graphs and networks quick notes

Notes on cheese-cracker