COGS-UN1001

concept-question

evergreen

formula

fruit

literature-note

Literature notes correspond with a single source (book, article, video, etc.). They will generally be formatted with the following sections:

• Summary – a paragraph to capture the main ideas from the text in an informational way, in the same vein as a scientific abstract. There may also be a link to a private “raw” note that stores bibliographic data and all my direct highlights.
• Takeaways or atomic notes – distinct from summaries, these are points that I want to apply directly to my life or thinking.
• Key terms – context-dependent definitions, often lifted directly from the source material. These may be linked to “definition” (permanent) notes that contain a slightly different operational definition, or link several variant definitions of the same term.
• Notes – bulleted, complete sentences that paraphrase the source. Significant or interesting points will be converted to atomic permanent notes.
  • Subheadings reflect subsections of the source material, whether explicitly or implicitly divided.
  • Direct quotes I find particularly interesting or informative may be copied over, but the bulk of text should be my own writing.
  • I will also write down questions, ephemeral observations, and possible connections to other permanent notes.

Making a literature note is my way of “processing” raw source material. The goal of processing is to incorporate what I read into my existing knowledge base in a concrete way. Ideally, processing is done at least one day after finishing the original source.

MATH-42X

MATH-GU4041

MATH-GU4042

MATH-GU4051

MATH-GU4053

MATH-GU4061

Overview

Columbia University, Fall 2024 – S. Hirsch

Course description

Real numbers, metric spaces, elements of general topology, sequences and series, continuity, differentiation, integration, uniform convergence, Ascoli-Arzela theorem, Stone-Weierstrass theorem.

Study status

TABLE WITHOUT ID
file.link as "Name",
lastmod as "Last Reviewed",
status as "Status"
 
FROM #MATH-GU4061
SORT lastmod ASC

Topics

The real and complex numbers

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Basic topology

Metric spaces

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Compactness

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Theorem: Interval nesting

Let $\{I_j{\}}_{j\in J}\subseteq \mathbb{R}$ be a collection of closed intervals such that $I_{j+1}\subseteq I_j$, and all $I_j$ are nonempty. Then their arbitrary intersection is nonempty, i.e., ${\bigcap }_{j=1}^{\infty }{I}_j\ne \varnothing$.

wip Why?

Theorem: Compactness of intervals in $\mathbb{R}$

Any closed interval $[a,b]\subseteq \mathbb{R}$ is compact, meaning there exists a finite subcover of $[a,b]$.

Theorem: Boxes in R^n are compact

Theorem: Cantor intersection

Let $A_i\subseteq {\mathbb{R}}^n$ be a family of compact sets such that the intersection of any finite collection of $A_i$ is nonempty. Then their arbitrary intersection is nonempty, i.e., ${\bigcap }_{i=1}^{\infty }{A}_i\ne \varnothing$.

(Path-)connectedness

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Bounded sets and functions

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Sequences and series

Sequences

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Cauchy sequences and complete metric spaces

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Limit point and sequential compactness

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Series

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Continuous functions

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(Path-)connectedness

Differentiable functions

Derivatives of real functions

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Local extrema of real functions

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Higher-order derivatives of real functions

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Integration

Development of the Darboux integral

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Darboux and Riemann integration

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Sequences of functions

Convergent sequences of functions

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Bounded sequences of functions and equicontinuity

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MATH-UN1207

MATH-UN1208

moc

open-question

permanent-note

Permanent notes are a sort of catch-all for any information I want to preserve that is not my original thinking (though the line is sometimes pretty fuzzy!). My notion of permanent notes is heavily inspired by Andy Matuschak’s evergreen notes.

Most permanent notes are “atomic” concepts (these are always titled with a complete proposition).

PHIL-UN2101

POLS-UN1501

polymetamath

pruned

seed

Seeds are my personal aggregates of multiple atomic notes or ideas. They contain the beginnings of original thinking.

STAT-GU4203

STAT-UN1201

topic-cognitive-science

topic-humanities

topic-information-computation-statistics

topic-logic-mathematics

topic-physics-complexity

wip