From 0c698311d5fe735f2c9d7908b5279c117cc4b977 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Fri, 1 Mar 2024 07:12:43 -0700 Subject: [PATCH] Notes on sequences. --- .../plugins/obsidian-to-anki-plugin/data.json | 9 +- notes/_journal/{ => 2024-02}/2024-02-29.md | 0 notes/_journal/2024-03-01.md | 11 ++ notes/algebra/sequences.md | 174 ++++++++++++++++++ notes/algorithms/sorting/merge-sort.md | 2 +- notes/logic/equiv-trans.md | 2 +- 6 files changed, 193 insertions(+), 5 deletions(-) rename notes/_journal/{ => 2024-02}/2024-02-29.md (100%) create mode 100644 notes/_journal/2024-03-01.md create mode 100644 notes/algebra/sequences.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 3c4a5c0..b55222a 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -122,7 +122,7 @@ "_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180", "_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb", "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970", - "logic/equiv-trans.md": "1f2d5823f128b8fdb526436e83a206ff", + "logic/equiv-trans.md": "315997b803b0748f5fc76d912ad97cda", "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889", "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2", "algorithms/loop-invariant.md": "29f9f9090a3109890d333a78acc18b50", @@ -199,7 +199,7 @@ "c17/index.md": "78576ee41d0185df82c59999142f4edb", "c17/escape-sequences.md": "ebc63c6cdfbe60bbc2708c1b0c8da8bb", "c17/declarations.md": "20e200f2b7abcab8f873cd080f4c9770", - "algorithms/sorting/merge-sort.md": "f66f482e5bd551c765fcba564c938d67", + "algorithms/sorting/merge-sort.md": "0a204db7441c761c6ee13689113760dd", "_journal/2024-02-24.md": "9bb319d5014caf962a9ce3141076cff4", "_journal/2024-02/2024-02-23.md": "0aad297148e8cc4058b48b7e45787ca7", "_journal/2024-02-25.md": "fb1a48208c11d12262facc647749ca6f", @@ -216,7 +216,10 @@ "git/index.md": "83d2d95fc549d9e8436946c7bd058d15", "encoding/integer.md": "5dd3a961bf259e53da207209051b8335", "_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508", - "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82" + "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82", + "_journal/2024-03-01.md": "e5602f6d41ae9dde85385428ad637d5a", + "_journal/2024-02/2024-02-29.md": "0e502a2c8baf90c2f12859b03f10b5a1", + "algebra/sequences.md": "97c217823aacf8910a1a37bde694ecfe" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-02-29.md b/notes/_journal/2024-02/2024-02-29.md similarity index 100% rename from notes/_journal/2024-02-29.md rename to notes/_journal/2024-02/2024-02-29.md diff --git a/notes/_journal/2024-03-01.md b/notes/_journal/2024-03-01.md new file mode 100644 index 0000000..7ed19b0 --- /dev/null +++ b/notes/_journal/2024-03-01.md @@ -0,0 +1,11 @@ +--- +title: "2024-03-01" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [ ] Interview Prep (1 Practice Problem) +- [ ] Log Work Hours (Max 3 hours) \ No newline at end of file diff --git a/notes/algebra/sequences.md b/notes/algebra/sequences.md new file mode 100644 index 0000000..6839b2d --- /dev/null +++ b/notes/algebra/sequences.md @@ -0,0 +1,174 @@ +--- +title: Sequences +TARGET DECK: Obsidian::STEM +FILE TAGS: algebra +tags: + - algebra +--- + +## Overview + +A **sequence** is an ordered list of numbers. These are usually represented as a **closed formula** or a **recursive definition**. + +%%ANKI +Basic +What is a sequence? +Back: An ordered list of values. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How does sequence $0, 1, 2, \ldots$ differ from set $\mathbb{N}$? +Back: Order matters in the former. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How is sequence $a_1, a_2, \ldots$ denoted? +Back: $(a_n)_{n \geq 1}$ +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What is the first index of $(a_n)_{n \geq 1}$? +Back: $1$ +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What terms make up the expanded form of sequence $(a_n)_{n \geq 0}$? +Back: $a_0, a_1, a_2, \ldots$ +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What term refers to the subscripts in sequence $a_0, a_1, a_2, \ldots$? +Back: Indices. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How do functions and sequences relate to one another? +Back: Every sequence is equivalent to a function mapping index to value. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How is sequence $a_0, a_1, a_2, \ldots$ interpreted as a function? +Back: As a function that assigns $n \mapsto a_n$. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What two methods allow exact specification of a sequence? +Back: Closed formulas and recursive definitions. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What is a closed formula for sequence e.g. $(a_n)_{n \geq 0}$? +Back: A formula that can be computed using a fixed number of "well-known" standard operations, independent of $n$. +Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994). + +END%% + +%%ANKI +Cloze +{$n!$} is a closed formula even though equivalent expression {$\Pi_{k=1}^n k$} is not. +Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994). + +END%% + +%%ANKI +Basic +Why is $\Pi_{k=1}^n k$ not considered a closed formula? +Back: The number of "well-known" standard operations (i.e. multiplications) scale with $n$. +Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994). + +END%% + +%%ANKI +Basic +Why is $n!$ considered a closed formula? +Back: Because the factorial operation is considered a "well-known" standard operation. +Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994). + +END%% + +%%ANKI +Basic +What two parts make up a recursive definition of a sequence? +Back: The recurrence relation and an initial condition. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What distinguishes a recurrence relation and a recursive definition? +Back: A recursive definition includes a recurrence relation along with an initial condition. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a closed formula? +Back: $a_n = 2^n$ +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a recursive definition? +Back: $a_n = 2 \cdot a_{n-1}$ with $a_0 = 1$. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Cloze +It is harder to find {terms of a sequence} with a recursive definition than closed formula. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +Which is considered easier to find: closed formulas or recursive definitions? +Back: Recursive definitions. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What general strategy is typically employed when finding a closed formula for a sequence? +Back: First find a recursive definition. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +## References + +* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). +* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994). diff --git a/notes/algorithms/sorting/merge-sort.md b/notes/algorithms/sorting/merge-sort.md index 01e1412..e2a6c10 100644 --- a/notes/algorithms/sorting/merge-sort.md +++ b/notes/algorithms/sorting/merge-sort.md @@ -71,7 +71,7 @@ END%% %%ANKI Basic What is the "combine" step of merge sort? -Back: Merge the subarrays sorted after the "conquer" step. +Back: Merge the sorted subarrays produced by the "conquer" step. Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). END%% diff --git a/notes/logic/equiv-trans.md b/notes/logic/equiv-trans.md index e2017d4..0375d8c 100644 --- a/notes/logic/equiv-trans.md +++ b/notes/logic/equiv-trans.md @@ -707,7 +707,7 @@ END%% %%ANKI Basic -What is the role of $\bar{e}$ role in textual substitution $E_{\bar{e}}^{\bar{x}}$? +What is the role of $\bar{e}$ in textual substitution $E_{\bar{e}}^{\bar{x}}$? Back: It is the expressions that are substituted into $E$. Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.