Link to previous part ROUGH NOTES (!) Updated: 11/10/24 Regular points; Manifolds; Tangents and Normals Back to top. \[{ \underline{\textbf{Regular points}} }\] Obs [Implicit function theorem for ${ C ^p }$ maps ${ \mathbb{R} ^n \times \mathbb{R} ^m \to \mathbb{R} ^m }$]: Let ${ f : U (\subseteq \mathbb{R} ^n \times \mathbb{R} ^m \te... Read more 11 Oct 2024 - 51 minute read
Link to previous part ROUGH NOTES (!) Updated: 12/7/24 Second derivative; Third derivative; Higher derivatives; Taylor’s theorem; Inverse function theorem; Implicit function theorem Back to top. \[{ \underline{\textbf{Second derivative}} }\] Def [${ C ^2 }$ maps]: Let ${ E, F }$ be complete normed spaces, and ${ f : U (\subseteq E \text... Read more 31 May 2024 - 84 minute read
Link to previous part ROUGH NOTES (!) Updated: 26/6/24 Integration of maps ${ f : [a, b] \to E }$; Mean value theorem Back to top. \[{ \underline{\textbf{Integration of maps } f : [a, b] \to E} }\] Recall if a map is uniformly continuous and into a complete space, the domain of definition can be expanded to its closure while preserving ... Read more 31 May 2024 - 46 minute read
\[{ \underline{\textbf{References}} }\] “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: 26/6/24 Continuous linear maps; Differe... Read more 31 May 2024 - 34 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