Lectures related to Advanced Counting: Generating Functions and Recurrence Relations

Counting with Recurrence Relations: Summary

Explores counting with recurrence relations, linear recurrence relations, generating functions, and counting principles.

Generating Functions: Advanced Counting

Covers the definition of generating functions and their application in solving recurrence relations.

Advanced Counting: Examples

Covers advanced counting techniques, including linear recurrence relations and generating functions, with examples from the Fibonacci sequence and differences between dice and poker cards.

Advanced Counting: Linear Homogeneous Recurrence Relations and Generating Functions

Explores solving linear homogeneous recurrence relations and generating functions for sequence formulas.

Linear Recurrence Relations

Explores linear recurrence relations, including examples like the Fibonacci numbers and the proof of related theorems.

Generating Functions for Correlation Functions

Explores generating functions for correlation functions in the context of the forced harmonic oscillator.

Fibonacci numbers and the golden ratio

Explores the Fibonacci sequence, identities, golden ratio, and explicit formulas.

Extended Binomial Theorem

Explores the Extended Binomial Theorem and counting problems using generating functions.

Ramanujan Graphs: Generating Functions and Expander Graphs

Explores Ramanujan graphs, generating functions, non-backtracking walks, and expander graphs in relation to NP-hard problems.

Generating Functions: Properties and Applications

Explores generating functions, Laplace transform, and their role in probability distributions.

Functions: Limits, Continuity, Differentiability

Explores the origin of functions, continuity, differentiability, and the physical representation of chemical bonds.

Linear Combinations: Moment-Generating Functions

Explores moment-generating functions, linear combinations, and normality of random variables.

Pentagonal Number Theorem and Jacobi Identity

Covers the pentagonal number theorem, Jacobi identity, and modularity of eta and theta functions.

Extreme Value Models: Technical Derivation

Explores the technical derivation and properties of Multivariate Extreme Value models.

Generating Functions: Moments and Cumulants

Explores generating functions for moments and cumulants, showcasing their role in distribution analysis.

Meromorphic Functions & Differentials

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Derivatives and Continuity in Multivariable Functions

Covers derivatives and continuity in multivariable functions, emphasizing the importance of partial derivatives.

Thermal Partition Function: Feynman Diagrams

Covers the thermal partition function using Feynman diagrams and perturbation theory.

Linear Recurrence Relations: Solving Techniques and Examples

Explains linear homogeneous recurrence relations and provides solving techniques and examples.

Functions Generators

Covers moment and cumulant generating functions for random variables, enabling calculation of moments and cumulants.