\[{ \underline{\textbf{References}} }\] “Undergraduate Analysis” by Lang. “Foundations of Applied Mathematics, Vol 1” by Humpherys, Jarvis, Evans. “Analysis Vol 2” by Amann, Escher. “Calculus on Normed Vector Spaces” by Coleman. “Differential Calculus” by Cartan. ROUGH NOTES (!) Updated: 16/2/25 Continuous linear maps; Differe... Read more 31 May 2024 - 35 minute read
Book: “Inference and Learning from Data” by Ali Sayed (Vols 1-3) Python code: Link ROUGH NOTES (!) Updated: 3/5/24 The books are organised as below. Ch-1 Matrix Theory: Code from booksite: None Thm: For a real symmetric matrix ${ A \in \mathbb{R} ^{N \times N}, }$ its eigenvalues (i.e. roots of ${ f(t) = \det(tI - A) }$) are all real. (... Read more 24 Apr 2024 - 70 minute read
Link to lectures Instructor: Prof. Jonathan Katz Book: “Introduction to Modern Cryptography” by Katz and Lindell ROUGH NOTES (!) Updated: 5/3/24 Week-1: Classical Cryptography focuses exclusively on ensuring secret communication between ${ 2 }$ parties sharing secret information in advance (That is, the focus is on private-key encryption s... Read more 16 Feb 2024 - 49 minute read
Link to Lectures Instructor: Prof. Michael Sipser Book: “Introduction to the Theory of Computation” by Michael Sipser ROUGH NOTES (!) Updated: 8/2/24 Lec-1: Computability Theory (1930s - 50s): What is computable or not ? Eg: Program verification, Pure mathematical truth Models of computation: Finite automata, Turing machines Comple... Read more 04 Feb 2024 - 27 minute read
Thm: Let ${ X }$ be a set with ${ \sigma -}$algebra ${ \mathfrak{M} }.$ Let ${ f : X \to [0, \infty] }$ be a measurable map. Then there exist simple measurable maps ${ 0 \leq s _1 \leq s _2 \leq \ldots (\leq f) }$ with pointwise limit ${ \lim _{n \to \infty} s _n (x) = f(x) }$ for all ${ x \in X }.$ Pf: We can try forming a sequence ${ \varphi... Read more 12 Dec 2023 - 1 minute read