Advanced Mathematics for Computer Science HNCST409 KTU BTech Honors 2024 Scheme

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Syllabus

Advanced Mathematics for Computer Science HNCST409 BTech Honors 2024 Model Question Paper and Answers

Module- I Number Theory 

Introduction

Group , Ring and Fields

Divisibility

Modular Arithmetic and Congruences

Equivalence Relation - Congruence

Linear Congruences

Solving simultaneous congruences- CRT

Modular Inverses 

   Euclidean algorithm 

   Extended Euclidean algorithm 

Euler’s and Fermat’s little theorem 

Euler's Totient Function

Prime Numbers and Prime-Power Factorization
Fermat and Mersenne Prime
Primality Testing and Factorization

Wilson's Theorem

Pseudo Primes and Carmichael Numbers

Primality testing 

    Miller-Rabin

    AKS

Galois Field (introduction and basic operations)

RSA cryptosystem: key generation - encryption/decryption 

Hash functions (introduction)

Module - II Optimization

Convex and Non Convex sets

Convex Hull

Identifying convex sets

Application of convex sets

Convex and Concave Functions

Identifying Convex and Concave Functions - Problems

Optimization Basics - Local Minima and Global Minima

Saddle Point

Review of Multi variate calculus

Gradient Based Method for Optimization - Gradient Descent Algorithm

Gradient Descent Example

Learning Rate (Step Size) in Optimization

Role of Gradient Descent in Machine Learning Algorithms

Stochastic Gradient Descent (SGD) and Mini-Batch Gradient Descent

Convergence Behavior of SGD and Batch Gradient Descent

Linear Regression using Gradient Descent

Module- III Bayesian Paradigm

Introduction to Probability

Bayes Theorem

Random Variables and Distribution Functions

Bayesian Paradigm: Probability as Belief

Conjugate Prior

Beta Distribution 

Bayesian inference for a coin toss (Beta prior + Binomial likelihood)

Beta–Binomial Model

Normal–Normal Bayesian Model

Poisson Distribution

Gamma Distribution

Conjugate Prior for Poisson Distribution

Introduction to Markov Chain Monte Carlo ( MCMC)

How MCMC Sampling Helps Approximate Complex Posterior Distributions

MCMC in Bayesian Estimation

Markov Decision Process (MDP) in Reinforcement Learning

Module- IV Graph Algorithms and Information Theory

Introduction to Graph and Definitions

Types of Graphs

Graph Representations

Dijikstra's Algorithm For Shortest Path in weighted Graph

Flow Graph -Max Flow Problem and Min Cut

Information Theory and Entropy

Cross Entropy

KL Divergence (Kullback–Leibler Divergence)

Difference Between KL Divergence and Cross-Entropy 

Page Rank Algorithm