diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 0d4d8d0..3518b2d 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -235,7 +235,29 @@ "dfs-edge-classification.png", "complex-plane-point.png", "nfa-example.png", - "topological-sort.png" + "topological-sort.png", + "unit-circle.png", + "circle-right.png", + "circle-left-down.png", + "abs-right.png", + "abs-left.png", + "abs-up.png", + "abs-down.png", + "abs-right-down.png", + "abs-left-down.png", + "circle-left-up.png", + "unit-circle-1-0.png", + "unit-circle-0-1.png", + "unit-circle-n1-0.png", + "unit-circle-0-n1.png", + "iterative-command.png", + "function-bijective.png", + "function-injective.png", + "function-surjective.png", + "function-general.png", + "function-kernel.png", + "triangular-gnomon.png", + "pascals-triangle.png" ], "File Hashes": { "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8", @@ -342,7 +364,7 @@ "combinatorics/inclusion-exclusion.md": "c27b49ee03cc5ee854d0e8bd12a1d505", "_journal/2024-02-21.md": "b9d944ecebe625da5dd72aeea6a916a2", "_journal/2024-02/2024-02-20.md": "af2ef10727726200c4defe2eafc7d841", - "algebra/radices.md": "474178afb07f3d5037c1547cc1a132f2", + "algebra/radices.md": "01fbaba8f81929581707d8df5ad0d912", "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a", "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7", "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd", @@ -375,7 +397,7 @@ "algebra/sequences/index.md": "7368b87313ea161a2655be0c39e705a3", "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", - "algebra/sequences/triangular-numbers.md": "aafaf54e5aff9ca3c7354591fce9f833", + "algebra/sequences/triangular-numbers.md": "12c81fa4d79d67c4853efcbc7b26f4c8", "algebra/sequences/square-numbers.md": "171f7c5a8dac088afba40923ab86c68e", "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670", "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2", @@ -538,7 +560,7 @@ "_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b", "_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365", "git/remotes.md": "6fbbc95efa421c720e40500e5d133639", - "programming/pred-trans.md": "5b271eebe32e33108d7a36ad98600148", + "programming/pred-trans.md": "007ac23931f767c84c5979ec83e28989", "set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7", "_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5", "_journal/2024-05/2024-05-13.md": "d549dd75fb42b4280d4914781edb0113", @@ -607,7 +629,7 @@ "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", "_journal/2024-06/2024-06-07.md": "c6bfc4c1e5913d23ea7828a23340e7d3", "lambda-calculus/alpha-conversion.md": "a68f3cc1565fb26335218986808a1190", - "lambda-calculus/index.md": "14bf297d4314414723c11a11211b35b5", + "lambda-calculus/index.md": "d68f65313a62110c5afa668b282149f3", "x86-64/instructions/condition-codes.md": "9c05ed99f5c96162e25f0ec4db55c656", "x86-64/instructions/logical.md": "a15c7da43cb97badef8ba4f8aadf9cbb", "x86-64/instructions/arithmetic.md": "e2c4c9caa51e089e313d6c9d3c3c0a12", @@ -627,7 +649,7 @@ "_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e", "_journal/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", "_journal/2024-06/2024-06-13.md": "e2722a00585d94794a089e8035e05728", - "set/functions.md": "bd4fbd92ac87631ba26637ac812b218b", + "set/functions.md": "a8f7fd819c27cdde6202da30787ea44c", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "8bbe0e1ca371756b91eec66af73911ce", "lambda-calculus/beta-reduction.md": "0935987f2bac0e6298735f2b26fd5885", @@ -912,7 +934,7 @@ "_journal/2024-10/2024-10-16.md": "cd778e1be2737462d885ae038c7b9744", "_journal/2024-10/2024-10-15.md": "c21679bd2c3b29f5a86d56a1fd23b18f", "_journal/2024-10-22.md": "4af65962007cfecdb2c679b44b56d25f", - "algorithms/dfs.md": "12a95fbc2fafaf87ee648c480ee041c3", + "algorithms/dfs.md": "aa499369c42a85c21861954b389a5819", "_journal/2024-10/2024-10-21.md": "de1a0861e87df29aeff11a291f8fbd45", "_journal/2024-10-23.md": "51b2ca6edf23b6a64fd7d3638a0b54cb", "_journal/2024-10/2024-10-22.md": "5ff4eb7eba58e77c4fb65b7162a485e6", @@ -944,7 +966,7 @@ "_journal/2024-11/2024-11-08.md": "806bbade5f8339579287687f9433334e", "_journal/2024-11/2024-11-07.md": "434ec3f15d7065ea740127aa8477dd17", "x86-64/directives.md": "019c1c1d04efb26c3e8758aac4543cc7", - "geometry/cartesian.md": "102453159fdb8525118d3995a132c997", + "geometry/cartesian.md": "68281a73f3949db43ad6a54e3e8c5cc2", "geometry/index.md": "cac68c1b624dbb0552e56cce47bcc21d", "_journal/2024-11-10.md": "5478337fd2017b99d0b359713a511e66", "_journal/2024-11/2024-11-09.md": "46f3a640223ef533f4523837b67b57c3", @@ -1003,7 +1025,7 @@ "_journal/2024-12-08.md": "5662897539b222db1af45dcd217f0796", "_journal/2024-12/2024-12-07.md": "bfb6c4db0acbacba19f03a04ec29fa5c", "linkers/static.md": "cc56ddfc33f605d26b954ec242abc4cf", - "linkers/index.md": "c6c2af6aab2773054b394c624bd2ddb6", + "linkers/index.md": "80e418eac44ad6e7d8bee799c7b11b18", "_journal/2024-12-09.md": "8988f0e8f0060f4b86d17e0bc4e7ff7e", "_journal/2024-12/2024-12-08.md": "5662897539b222db1af45dcd217f0796", "_journal/2024-12-10.md": "c12d380d24d7d1dc2e74a57a1b79399e", @@ -1049,7 +1071,7 @@ "_journal/2024-12/2024-12-21.md": "1c1a5791f7519c92e882957cf417b51f", "formal-system/language.md": "7797d33a0b0eb187d43dda46a138fb25", "computability/automaton.md": "1dd5048ea2a66d8090a85945593fcf68", - "computability/index.md": "d7938428ed0b0224c1fe1e59d1fab118", + "computability/index.md": "16ae7a270363055f7096b7dd8c09a977", "_journal/2024-12-23.md": "72b0964a8a5ed8ba0acf7fe10b5de279", "_journal/2024-12/2024-12-22.md": "75375a867efc5b3aff406c73394d4814", "computability/language.md": "9ee8bd16c231e71855ab1d8dae3188cb", @@ -1059,13 +1081,13 @@ "_journal/2024-12/2024-12-23.md": "72b0964a8a5ed8ba0acf7fe10b5de279", "_journal/2024-12-25.md": "1717d37b074df58175ec0272adc278de", "_journal/2024-12/2024-12-24.md": "dcd3bd8b82ca4d47a9642a46d8bece0d", - "linkers/relocatable.md": "b6f0c13e07ed57ea73dea6b4a72560d1", + "linkers/relocatable.md": "ac24efbabe07222a89acb2fd5135cdb3", "data-models/federation.md": "1d92747304186bd2833a00a488fcac48", "_journal/2024-12-26.md": "022aeaf68d46fd39b23aca9c577f3f41", "_journal/2024-12/2024-12-25.md": "1717d37b074df58175ec0272adc278de", "_journal/2024-12-27.md": "abc4a39a50305f3558181189eefb2058", "_journal/2024-12/2024-12-26.md": "59e59cad1ae568adbe8e27e98d36c59c", - "combinators/index.md": "8e324bbcf49cca9c0c0f9bbf843cbebb", + "combinators/index.md": "8b55b44c955da88368b0b5635909b064", "_journal/2024-12-28.md": "1ad3caec4ea6f597cc5156f19b274c50", "_journal/2024-12/2024-12-27.md": "abc4a39a50305f3558181189eefb2058", "_journal/2024-12-29.md": "e7808872f56a12b51165fc86a1c48e60", @@ -1425,7 +1447,7 @@ "_journal/2025-01/2025-01-06.md": "20030a4b6a1f8f4b2cb882c6d4c59f29", "_journal/2025-01/2025-01-05.md": "0217401ed8718d4354d856a92a19a345", "_journal/2025-01/2025-01-07.md": "bd5aa36eff9211a9a89cc47f1c2dbdcd", - "data-models/rdf/rdfs.md": "f891b5385d3f41acc0c6a8ce88186419", + "data-models/rdf/rdfs.md": "705777f026ea8bf4d8311f1c47621cd4", "_journal/2025-01-11.md": "a9bdad00db9432ea97df265bca1f8261", "_journal/2025-01/2025-01-10.md": "a7929276f89cc19193622dd1f1dd2588", "_journal/2025-01/2025-01-09.md": "166ff75c5ea1bf5110931fa054e1565e", @@ -1440,8 +1462,14 @@ "_journal/2025-01/2025-01-14.md": "88eb99d4319693c7f4cd2357618a19f8", "_journal/2025-01/2025-01-15.md": "a559a6eba2958e2664ad25c1e3236d87", "_journal/2025-01-16.md": "e3a21059205784a4e88bfe3b4deac7f7", - "_journal/2025-01-17.md": "ba60278a6cca1832ad28c273b01b0745", - "_journal/2025-01/2025-01-16.md": "e3a21059205784a4e88bfe3b4deac7f7" + "_journal/2025-01-17.md": "08a5f05bb572db9495bfc2b4feb8e0a9", + "_journal/2025-01/2025-01-16.md": "e3a21059205784a4e88bfe3b4deac7f7", + "trigonometry/index.md": "6d07d5ba5e352f182a19b8a3d804321b", + "geometry/circle.md": "751a5aada6521281f1d8edb463572943", + "_journal/2025-01-18.md": "8655fbf94aeec13efe9b6d2087c1f37e", + "_journal/2025-01/2025-01-17.md": "08a5f05bb572db9495bfc2b4feb8e0a9", + "_journal/2025-01-19.md": "a37c6f534cf5e272619c5f813974afcf", + "_journal/2025-01/2025-01-18.md": "7a1655887093f37ffe86309d90459b3b" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2025-01-17.md b/notes/_journal/2025-01-17.md deleted file mode 100644 index ce35974..0000000 --- a/notes/_journal/2025-01-17.md +++ /dev/null @@ -1,9 +0,0 @@ ---- -title: "2025-01-17" ---- - -- [ ] Anki Flashcards -- [x] KoL -- [x] OGS -- [ ] Sheet Music (10 min.) -- [ ] Korean (Read 1 Story) \ No newline at end of file diff --git a/notes/_journal/2025-01-19.md b/notes/_journal/2025-01-19.md new file mode 100644 index 0000000..ef1a9c3 --- /dev/null +++ b/notes/_journal/2025-01-19.md @@ -0,0 +1,11 @@ +--- +title: "2025-01-19" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* More notes on relocation entries. \ No newline at end of file diff --git a/notes/_journal/2025-01/2025-01-17.md b/notes/_journal/2025-01/2025-01-17.md new file mode 100644 index 0000000..a5a0c6e --- /dev/null +++ b/notes/_journal/2025-01/2025-01-17.md @@ -0,0 +1,13 @@ +--- +title: "2025-01-17" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Distinguish asserted and inferred RDF triples. +* Notes on the [[circle]] and [[trigonometry/index#Unit Circle|unit circle]]. +* Formal definition of the [[pred-trans#Iterative|iterative]] command. \ No newline at end of file diff --git a/notes/_journal/2025-01/2025-01-18.md b/notes/_journal/2025-01/2025-01-18.md new file mode 100644 index 0000000..151acd0 --- /dev/null +++ b/notes/_journal/2025-01/2025-01-18.md @@ -0,0 +1,11 @@ +--- +title: "2025-01-18" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on [[relocatable#Relocation Entries|relocation entries]]. \ No newline at end of file diff --git a/notes/algebra/radices.md b/notes/algebra/radices.md index b30b4aa..472e1be 100644 --- a/notes/algebra/radices.md +++ b/notes/algebra/radices.md @@ -202,6 +202,33 @@ Tags: binary::hex END%% +%%ANKI +Basic +What hexadecimal value does $2^{0 + 4(0)}$ evaluate to? +Back: `0x1` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +What hexadecimal value does $2^{0 + 4(2)}$ evaluate to? +Back: `0x100` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +Write `0x10000` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this? +Back: $i = 0$ and $j = 4$. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + %%ANKI Basic How is the $1$ in $2^{1 + 4j}$ translated to hex? @@ -211,6 +238,33 @@ Tags: binary::hex END%% +%%ANKI +Basic +What hexadecimal value does $2^{1 + 4(2)}$ evaluate to? +Back: `0x200` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +What hexadecimal value does $2^{1 + 4(3)}$ evaluate to? +Back: `0x2000` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +Write `0x200` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this? +Back: $i = 1$ and $j = 2$. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + %%ANKI Basic How is the $2$ (power) in $2^{2 + 4j}$ translated to hex? @@ -220,6 +274,24 @@ Tags: binary::hex END%% +%%ANKI +Basic +What hexadecimal value does $2^{2 + 4(1)}$ evaluate to? +Back: `0x40` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +Write `0x4000` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this? +Back: $i = 2$ and $j = 3$. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + %%ANKI Basic How is the $3$ in $2^{3 + 4j}$ translated to hex? @@ -229,6 +301,24 @@ Tags: binary::hex END%% +%%ANKI +Basic +What hexadecimal value does $2^{3 + 4(0)}$ evaluate to? +Back: `0x8` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + +%%ANKI +Basic +Write `0x80` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this? +Back: $i = 3$ and $j = 1$. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: binary::hex + +END%% + %%ANKI Basic How is $n$ in $2^n$ factored to quickly write the decimal value's hex representation? diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md index b65cbcc..bc2d60c 100644 --- a/notes/algebra/sequences/triangular-numbers.md +++ b/notes/algebra/sequences/triangular-numbers.md @@ -15,7 +15,7 @@ The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of wh Basic What is a polygonal number? Back: A number of pebbles that can be arranged into the shape of a regular filled polygon. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -23,7 +23,7 @@ END%% Basic What is a figurate number? Back: Polygonal numbers and their generalizations to other dimensions. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -31,7 +31,7 @@ END%% Basic What are considered the simplest polygonal numbers? Back: The triangular numbers. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -39,7 +39,7 @@ END%% Basic How do polygonal numbers relate to figurate numbers? Back: Polygonal numbers are a subset of the figurate numbers. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -47,7 +47,7 @@ END%% Basic What is a gnomon? Back: The "piece" added to a figurate number to transform it to the next larger one. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -55,7 +55,7 @@ END%% Basic What shape do gnomons associated with triangular numbers take on? Back: Lines. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -64,7 +64,7 @@ Basic How are gnomons of the triangular numbers visualized? Back: ![[triangular-gnomon.png]] -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -73,7 +73,7 @@ Basic What general term refers to the highlighted portion of pebbles in the following? ![[triangular-gnomon.png]] Back: Gnomons. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -81,15 +81,24 @@ END%% Basic The triangular numbers correspond to what kind of triangles? Back: Equilateral triangles. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% +%%ANKI +Basic +How do the triangular numbers correspond to *equilateral* triangles? +Back: +![[triangular-gnomon.png]] +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + %%ANKI Basic What is the first triangular *and* square number? Back: $36$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -97,7 +106,7 @@ END%% Basic What are the first five triangular numbers $(T_n)_{n \geq 0}$? Back: $0, 1, 3, 6, 10$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -111,7 +120,7 @@ Back: * * * * * * * ``` -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -119,7 +128,7 @@ END%% Basic How is the $n$th triangular number written as a summation? Back: $\sum_{k=1}^n k$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -127,7 +136,7 @@ END%% Basic What polygonal sequence is the summation analog of factorial? Back: The triangular numbers. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -135,7 +144,7 @@ END%% Basic What notation does Knuth introduce to denote the $n$th triangular number? Back: $n?$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -143,14 +152,14 @@ END%% Basic What name does Knuth give the LHS of $n? = \sum_{k=1}^n k$? Back: The termial. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%ANKI Cloze The {1:term}ial is to {2:$n?$} as the {2:factor}ial is to {1:$n!$}. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -158,7 +167,7 @@ END%% Basic What closed formula is traditionally used to compute the $n$th triangular number? Back: $\large{\frac{n(n + 1)}{2}}$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -166,7 +175,7 @@ END%% Basic What is the recurrence relation in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$? Back: $T_n = T_{n-1} + n$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -174,7 +183,7 @@ END%% Basic What is the initial condition(s) in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$? Back: $T_0 = 0$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -196,7 +205,7 @@ END%% Basic What combinatorial closed formula is used to compute the $n$th triangular number? Back: $\binom{n + 1}{2}$ -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -204,7 +213,7 @@ END%% Basic What is the combinatorial explanation as to why the $n$th triangular number is $\binom{n + 1}{2}$? Back: $\sum_{k=1}^n k$ is the number of ways distinct pairs can be made from $n + 1$ objects. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -213,7 +222,7 @@ Basic Where in Pascal's triangle are the natural numbers embedded? Back: Along the second leftward diagonal: ![[pascals-triangle.png]] -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -222,7 +231,7 @@ Basic Where in Pascal's triangle are the triangular numbers embedded? Back: Along the third leftward diagonal: ![[pascals-triangle.png]] -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -236,7 +245,7 @@ for (int i = 1; i <= n; ++i) { } ``` Back: The $n$th triangular number. -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -250,7 +259,7 @@ Back: $2 \cdot T_n$ is the number of units in an $n \times (n + 1)$ rectangle, e * * - - - * - - - - ``` -Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% @@ -265,4 +274,4 @@ END%% ## Bibliography * 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). -* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). \ No newline at end of file +* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). \ No newline at end of file diff --git a/notes/algorithms/dfs.md b/notes/algorithms/dfs.md index 1f9c573..5b8546b 100644 --- a/notes/algorithms/dfs.md +++ b/notes/algorithms/dfs.md @@ -534,6 +534,7 @@ END%% %%ANKI Basic +Why is edge $\langle b, s \rangle$ classified as a back edge? ![[dfs-edge-classification.png]] Back: Because $s$ is an ancestor of $b$. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). diff --git a/notes/combinators/index.md b/notes/combinators/index.md index ab48329..8f0a1b5 100644 --- a/notes/combinators/index.md +++ b/notes/combinators/index.md @@ -335,6 +335,77 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi END%% +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}\mathbf{S}))\mathbf{K})$ a combinator? +Back: N/A. It is. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}x))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a combinator? +Back: It contains atom $x$ which isn't a basic combinator. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}0))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a combinator? +Back: It contains atom $0$ which isn't a basic combinator. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}\mathbf{S}))\mathbf{K})$ a closed term? +Back: N/A. It is. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}x))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a closed term? +Back: It contains variable $x$. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}0))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a closed term? +Back: N/A. It is. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Cloze +By convention, parentheses in combinatory logic are {left}-associative. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +How is $CL$-term $UVWX$ written with parentheses reintroduced? +Back: $(((UV)W)X)$ +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +In combinatory logic, is $UVW \equiv ((UV)W)$? +Back: Yes. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + ## Basic Combinators The combinatory logic is a notation that eliminate the need for quantified variables. We start with basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. These **basic combinators** are defined as: diff --git a/notes/computability/index.md b/notes/computability/index.md index d0712b8..9138ca8 100644 --- a/notes/computability/index.md +++ b/notes/computability/index.md @@ -247,11 +247,10 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed END%% %%ANKI -Basic -A language is a set satisfying what? -Back: It contains strings over some alphabet. +Cloze +A {language} is a set containing {strings} over some {alphabet}. Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013). - + END%% %%ANKI diff --git a/notes/data-models/rdf/rdfs.md b/notes/data-models/rdf/rdfs.md index eced8e0..cd38ca7 100644 --- a/notes/data-models/rdf/rdfs.md +++ b/notes/data-models/rdf/rdfs.md @@ -8,7 +8,7 @@ tags: ## Overview -The simplest extension to RDF that allows a modeler to manage inference is **RDF Schema** (RDFS). +The simplest extension to RDF that allows a modeler to manage inference is **RDF Schema** (RDFS). Triples that are inserted directly into an underlying RDF store are called **asserted triples**. Triples that are derived from inference rules are called **inferred triples**. %%ANKI Basic @@ -26,6 +26,44 @@ Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web END%% +%%ANKI +Cloze +An {asserted} triple is contrary to an {inferred} triple. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What does it mean for a triple to be asserted? +Back: The triple exists directly in the underlying RDF store. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +What does it mean for a triple to be inferred? +Back: The triple is derived from some set of inference rules. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Cloze +{Asserted} triples are used to derive {inferred} triples. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + +%%ANKI +Basic +When might a triple be both asserted and inferred? +Back: When the inference engine infers an already existing triple. +Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020. + +END%% + ## Classes All things described by RDF are called **resources**. Resources may be divided into groups called **classes**. Classes are themselves resources. The `rdf:type` property may be used to state that a resource is an instance of a class. Associated with a class is a set, called the **class extension** of the class, which is the set of the instances of the class. A class may be a member of its own class extension. diff --git a/notes/geometry/cartesian.md b/notes/geometry/cartesian.md index 729977a..a78c2ca 100644 --- a/notes/geometry/cartesian.md +++ b/notes/geometry/cartesian.md @@ -213,22 +213,6 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces END%% -%%ANKI -Basic -Which of horizontal and/or vertical transformations "act inversely"? -Back: Horizontal transformations. -Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). - -END%% - -%%ANKI -Basic -Which of horizontal and/or vertical transformations "act normally"? -Back: Vertical transformations. -Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html). - -END%% - %%ANKI Basic How is the graph of $y = f(x)$ transformed in the graph of $y = \frac{1}{2}f(\frac{x}{3})$? diff --git a/notes/geometry/circle.md b/notes/geometry/circle.md new file mode 100644 index 0000000..28e99d3 --- /dev/null +++ b/notes/geometry/circle.md @@ -0,0 +1,200 @@ +--- +title: Circle +TARGET DECK: Obsidian::STEM +FILE TAGS: geometry::circle +tags: + - circle + - geometry +--- + +## Overview + +A **circle** is a shape consisting of all points in a plane at a given distance from a given point, i.e. the **center**. In this way, a circle is characterized by its center and **radius**, the distance from the center to any point of the circle. The perimeter of a circle is known as its **circumference**. + +On the [[cartesian|Cartesian coordinate system]], the equation of a circle with radius $r$ and center $\langle h, k \rangle$ is $$(x - h)^2 + (y - k)^2 = r^2.$$ + +Such a circle has circumference $2\pi r$ and [[area]] $\pi r^2$. + +%%ANKI +Basic +A circle is characterized by what two properties? +Back: Its center and its radius. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +In plain English, describe what a circle is. +Back: A set of points at a given distance from some given point. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +The perimeter of a circle is known as what? +Back: Its circumference. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +What is the Cartesian equation of a circle with radius $r$ and center $\langle h, k \rangle$? +Back: $(x - h)^2 + (y - k)^2 = r^2$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Given indeterminates $x$ and $y$, the following equation describes what shape? $$(x - h)^2 + (y - k)^2 = r^2$$ +Back: A circle with radius $r$ and center $\langle h, k \rangle$. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with radius $r$. What does its circumference evaluate to? +Back: $2 \pi r$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with radius $r$. What is its area? +Back: $\pi r^2$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with diameter $d$. What does its circumference evaluate to? +Back: $\pi d$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with diameter $d$. What its area? +Back: $\pi \left(\frac{d}{2}\right)^2$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with radius $r$. What does the following evaluate to? $$2 \pi r$$ +Back: Its circumference. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Consider a circle with radius $r$. What does the following evaluate to? $$\pi r^2$$ +Back: Its area. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +What is the Cartesian equation of the following shape? +![[unit-circle.png]] +Back: $x^2 + y^2 = 1$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +What is the Cartesian equation of the following shape? +![[circle-right.png]] +Back: $(x - 2)^2 + y^2 = 4$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +What is the Cartesian equation of the following shape? +![[circle-left-down.png]] +Back: $(x + 1)^2 + (y + 1)^2 = 4$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +What is the Cartesian equation of the following shape? +![[circle-left-up.png]] +Back: $(x + 1)^2 + (y - 1)^2 = 4$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Rewrite equation $x^2 + y^2 = 1$ shifted left by $a > 0$. +Back: $(x + a)^2 + y^2 = 1$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Rewrite equation $x^2 + y^2 = 1$ shifted up by $b > 0$. +Back: $x^2 + (y - b)^2 = 1$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Rewrite equation $x^2 + y^2 = 1$ shifted right by $a > 0$. +Back: $(x - a)^2 + y^2 = 1$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +Rewrite equation $x^2 + y^2 = 1$ shifted down by $b > 0$. +Back: $x^2 + (y + b)^2 = 1$ +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y - 3)^2 = 1$$ +Back: Right by $3$ and up by $3$. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +In what direction(s) is the unit circle shifted in the following? $$(x + 3)^2 + (y - 3)^2 = 1$$ +Back: Left by $3$ and up by $3$. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +%%ANKI +Basic +In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y + 3)^2 = 1$$ +Back: Right by $3$ and down by $3$. +Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). + +END%% + +## Bibliography + +* “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). \ No newline at end of file diff --git a/notes/geometry/images/circle-left-down.png b/notes/geometry/images/circle-left-down.png new file mode 100644 index 0000000..8546d24 Binary files /dev/null and b/notes/geometry/images/circle-left-down.png differ diff --git a/notes/geometry/images/circle-left-up.png b/notes/geometry/images/circle-left-up.png new file mode 100644 index 0000000..7c05486 Binary files /dev/null and b/notes/geometry/images/circle-left-up.png differ diff --git a/notes/geometry/images/circle-right.png b/notes/geometry/images/circle-right.png new file mode 100644 index 0000000..62fc7da Binary files /dev/null and b/notes/geometry/images/circle-right.png differ diff --git a/notes/geometry/images/unit-circle.png b/notes/geometry/images/unit-circle.png new file mode 100644 index 0000000..ea9d278 Binary files /dev/null and b/notes/geometry/images/unit-circle.png differ diff --git a/notes/lambda-calculus/index.md b/notes/lambda-calculus/index.md index 1dd3bee..38595d2 100644 --- a/notes/lambda-calculus/index.md +++ b/notes/lambda-calculus/index.md @@ -214,7 +214,7 @@ END%% %%ANKI Basic -How is expression $\lambda x. \lambda y. MN$ written with parentheses reintroduced? +How is $\lambda$-term $\lambda x. \lambda y. MN$ written with parentheses reintroduced? Back: $(\lambda x. (\lambda y. (MN)))$ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). diff --git a/notes/linkers/index.md b/notes/linkers/index.md index d22b9c8..7c34395 100644 --- a/notes/linkers/index.md +++ b/notes/linkers/index.md @@ -10,10 +10,8 @@ tags: To build an executable, a linker must perform two main tasks: -1. **Symbol resolution**. The linker must associate each symbol reference with exactly one symbol definition. -2. **Relocation**. The linker must relocate code and data sections by associating a memory location with each symbol definition, and then modifying all of the references to those symbols so that they point to this memory location. - -The linker blindly performs relocations using detailed instructions generated by the assembler called **relocation entries**. +1. **Symbol resolution**. The linker phase in which each symbol reference is associated with exactly one symbol definition. +2. **Relocation**. The linker phase in which code and data sections across input modules are combined. Each section, along with the symbols defined in them, are assigned unique run-time memory addresses. Additionally, references to symbols are updated so they point to the correct run-time addresses. %%ANKI Basic @@ -31,6 +29,14 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +%%ANKI +Basic +After which linker phase is the size of the code and data sections of each input object module known? +Back: Symbol resolution. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + %%ANKI Basic What is the goal of symbol resolution? @@ -39,6 +45,13 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +%%ANKI +Cloze +The {symbol resolution} phase associates each {symbol reference} with exactly one {symbol definition}. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + %%ANKI Basic What is the goal of relocation? @@ -55,13 +68,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% -%%ANKI -Cloze -The {symbol resolution} phase associates each {symbol reference} with exactly one {symbol definition}. -Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. - -END%% - %%ANKI Cloze The {relocation} phase assigns a {memory location} to each symbol and {updates references} accordingly. @@ -79,24 +85,25 @@ END%% %%ANKI Basic -What is emitted by the assembler to help the linker relocate sections? -Back: Relocation entries. +Which linker phase combines code and data sections of the same type into a new aggregate section? +Back: Relocation. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. - + +END%% + +%%ANKI +Basic +Which linker phase is responsible for building up the `.symtab` section? +Back: Symbol resolution. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + END%% %%ANKI Cloze -The assembler outputs {relocation entries} to guide the linker during {relocation}. +{Relocatable} object files are merged together to make an {executable} object file. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. - -END%% - -%%ANKI -Cloze -The {1:assembler} outputs relocation entries to guide the {1:linker} during relocation. -Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. - + END%% ## Object Files diff --git a/notes/linkers/relocatable.md b/notes/linkers/relocatable.md index d7d5290..fe1cb08 100644 --- a/notes/linkers/relocatable.md +++ b/notes/linkers/relocatable.md @@ -637,6 +637,335 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program END%% +## Relocation Entries + +Whenever the assembler encounters a reference to an object whose ultimate location is unknown, it generates a **relocation entry** that tells the linker how to modify the reference when it merges the object file into an executable. Each entry looks something like: + +```c +struct Elf64_Rela { + long offset; // Offset of the reference to relocate + long type : 32, // Relocation type + symbol : 32; // Symbol table index + long addend; // Additional constant used to bias the value +}; +``` + +%%ANKI +Basic +What is emitted by the assembler to help the linker relocate sections? +Back: Relocation entries. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +The assembler outputs {relocation entries} to guide the linker during {relocation}. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +The {1:assembler} outputs relocation entries to guide the {1:linker} during relocation. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Which component of a compiler driver produces relocation entries? +Back: The assembler. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +How many relocation entries are produced for any given object module? +Back: One for every reference whose ultimate address is unknown. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +*Why* aren't relocation entries relevant for executable object files? +Back: All memory addresses should be resolved once the executable is produced. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +Relocation entries for {data} are placed in {`.rel.data`}. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Cloze +Relocation entries for {code} are placed in {`.rel.text`}. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +At what point during linking are relocation entries no longer necessary? +Back: When the fully merged executable object file is produced. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +Relocation entries are included in what kind of object module? +Back: Relocatable object modules. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. + +END%% + +%%ANKI +Basic +A relocation entry corresponds to what kind of C construct? +Back: A `struct`. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What is the purpose of `offset`? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: It is the offset of the reference to relocate. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What is the `offset` measured relative to? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: The start of the section the reference is located in. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What is the purpose of `symbol`? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: It identifies the symbol the modified reference should point to. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. `symbol` is an index into what table? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `.symtab` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What is the purpose of `addend`? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: To bias the value of the modified reference. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What is the purpose of `type`? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: To indicate to the linker what algorithm should be used to compute the relocated address. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What are the two most basic values of `type`? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `R_X86_64_32` and `R_X86_64_PC32`. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Cloze +{1:`R_X86_64_32`} is to {2:absolute} whereas {2:`R_X86_64_PC32`} is to {1:relative}. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: linker::elf x86-64 + +END%% + +%%ANKI +Basic +What is the significance of `R` in type `R_X86_64_32`? +Back: It is the prefix used for relocation entry `type` values. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +What is the significance of `32` in type `R_X86_64_32`? +Back: The reference is relocated using a 32-bit absolute address. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +What is the significance of `PC` in type `R_X86_64_PC32`? +Back: It is short for **p**rogram **c**ounter. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +What is the significance of `PC32` in type `R_X86_64_PC32`? +Back: The reference is relocated using a 32-bit PC-relative address. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +`R_X86_64_32` is a possible value for what field in the following ELF relocation entry? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `type` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +`R_X86_64_PC32` is a possible value for what field in the following ELF relocation entry? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `type` +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What should `type` be if relocating PC-relatively? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `R_X86_64_PC32`. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + +%%ANKI +Basic +Consider the following ELF relocation entry. What should `type` be if relocating absolutely? +```c +struct Elf64_Rela { + long offset; + long type : 32; + long symbol : 32; + long addend; +}; +``` +Back: `R_X86_64_32`. +Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. +Tags: c17 linker::elf x86-64 + +END%% + ## Bibliography * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. \ No newline at end of file diff --git a/notes/programming/pred-trans.md b/notes/programming/pred-trans.md index 2496c83..7ab14c6 100644 --- a/notes/programming/pred-trans.md +++ b/notes/programming/pred-trans.md @@ -1035,6 +1035,9 @@ END%% The general form of the **iterative command** is: $$\begin{align*} \textbf{do } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{od } & \end{align*}$$ +We denote the iterative command as $\text{DO}$ and define $\text{DO}$ in terms of $wp$ as: $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +where $H_k$ is given [[algebra/sequences/index|recursive definition]]: $$\begin{align*} H_0(R) & = \neg (B_1 \lor \cdots \lor B_n) \land R \\ H_{k+1}(R) & = H_0(R) \lor wp(\text{IF}, H_k(R)) \end{align*}$$ + %%ANKI Basic The conventional `while` statement corresponds to what command? @@ -1123,6 +1126,130 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in END%% +%%ANKI +Cloze +{1:$\text{IF}$} is to {2:$abort$} whereas {2:$\text{DO}$} is to {1:$skip$}. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +Given associated recursive definition $H_k$, what is the formal definition of $\text{DO}$? +Back: For some predicate $R$, $wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +In the following definition, what does $H_k(R)$ represent? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +Back: The set of states in which execution of $\text{DO}$ terminates in $k$ or fewer iterations. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +In the following definition, how is $H_0$ defined? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +Back: Given guards $B_1, \ldots, B_n$, as $H_0 = \neg (B_1 \lor \cdots \lor B_n) \land R$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +In the following definition, what set of states does $H_0(R)$ correspond to? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +Back: Those in which $\text{DO}$ finishes execution in $0$ iterations with $R$ true. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +In the following definition, how is $H_{k+1}$ defined? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +Back: As $H_{k+1}(R) = H_0(R) \lor wp(\text{IF}, H_k(R))$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +In the following definition, what set of states does $H_k(R)$ correspond to? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$ +Back: Those in which $\text{DO}$ finishes execution in $k$ or fewer iterations with $R$ true. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +Let $H_k$ denote the associated recursive definition of $wp(\text{DO}, R)$. *Why* does $H_k \Rightarrow H_{k+1}$? +Back: $H_{k+1}$ is the set of states in which $\text{DO}$ finishes execution in $k$ *or fewer* iterations. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +Let $H_k$ denote the associated recursive definition of $wp(\text{DO}, R)$. *Why* does $H_{k + 1} \Rightarrow H_k$? +Back: N/A. It doesn't. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the associated recursive definition of $wp(\text{DO}, R)$ described in plain English? +Back: As the set of states in which execution of $\text{DO}$ terminates in a finite number of iterations with $R$ true. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Cloze +Iterative command {$\textbf{do od}$} is equivalent to command {$skip$}. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +*Why* does command $\textbf{do od}$ skip? +Back: The $\text{DO}$ command iterates until no guard is satisfied. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What does $wp(\textbf{do } T \rightarrow skip \textbf{ od}, R)$ evaluate to? +Back: $F$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +*Why* does $wp(\textbf{do } T \rightarrow skip \textbf{ od}, R)$ evaluate to $F$? +Back: $\textbf{do } T \rightarrow skip \text{ od}$ never terminates. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Cloze +The {$\text{DO}$} command corresponds to zero or more {$\text{IF}$} commands. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Cloze +Let $R$ be a predicate. Then $wp(\text{DO}, R) = \exists k,$ {$k \geq 0$} $\land$ $H_k(R)$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + ## Bibliography * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. \ No newline at end of file diff --git a/notes/set/functions.md b/notes/set/functions.md index fe4ea87..1878148 100644 --- a/notes/set/functions.md +++ b/notes/set/functions.md @@ -199,7 +199,7 @@ END%% %%ANKI Basic Let $A$ and $B$ be disjoint sets. $f \colon A \rightarrow B$ is an operation on what set? -Back: N/A. +Back: N/A. $f$ is not an operation. Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938). END%% diff --git a/notes/trigonometry/images/unit-circle-0-1.png b/notes/trigonometry/images/unit-circle-0-1.png new file mode 100644 index 0000000..09e1390 Binary files /dev/null and b/notes/trigonometry/images/unit-circle-0-1.png differ diff --git a/notes/trigonometry/images/unit-circle-0-n1.png b/notes/trigonometry/images/unit-circle-0-n1.png new file mode 100644 index 0000000..326c9f5 Binary files /dev/null and b/notes/trigonometry/images/unit-circle-0-n1.png differ diff --git a/notes/trigonometry/images/unit-circle-1-0.png b/notes/trigonometry/images/unit-circle-1-0.png new file mode 100644 index 0000000..b28f536 Binary files /dev/null and b/notes/trigonometry/images/unit-circle-1-0.png differ diff --git a/notes/trigonometry/images/unit-circle-n1-0.png b/notes/trigonometry/images/unit-circle-n1-0.png new file mode 100644 index 0000000..cd6fdae Binary files /dev/null and b/notes/trigonometry/images/unit-circle-n1-0.png differ diff --git a/notes/trigonometry/index.md b/notes/trigonometry/index.md new file mode 100644 index 0000000..41493ea --- /dev/null +++ b/notes/trigonometry/index.md @@ -0,0 +1,215 @@ +--- +title: Trigonometry +TARGET DECK: Obsidian::STEM +FILE TAGS: trigonometry +tags: + - trigonometry +--- + +## Overview + +Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has generalized to studying periodicity. + +%%ANKI +Basic +Trigonometry was originally the study of what geometric shape? +Back: Triangles. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +## Unit Circle + +On the [[cartesian|Cartesian coordinate system]], the **unit circle** is the [[circle]] with center at the origin and radius $1$. + +%%ANKI +Basic +On the Cartesian coordinate system, what is the unit circle? +Back: The circle with center at the origin and radius $1$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +On the Cartesian coordinate system, where is the center of the unit circle located? +Back: At $\langle 0, 0 \rangle$, i.e. the origin. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the radius of the unit circle? +Back: $1$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the diameter of the unit circle? +Back: $2$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the circumference of the unit circle? +Back: $2\pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the area of the unit circle? +Back: $\pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers does the point $\langle 0, 0 \rangle$ on the unit circle map to? +Back: N/A. This point is not on the circle itself. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers does the point $\langle 1, 0 \rangle$ on the unit circle map to? +Back: $2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which point on the unit circle does number $2\pi$ map to? +Back: $\langle 1, 0 \rangle$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which point on the unit circle does number $\frac{3\pi}{2}$ map to? +Back: $\langle 0, -1 \rangle$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers does the point $\langle 0, -1 \rangle$ on the unit circle map to? +Back:$\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers does the point $\langle 0, 1 \rangle$ on the unit circle map to? +Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which point on the unit circle does number $\frac{\pi}{2}$ map to? +Back: $\langle 0, 1 \rangle$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which point on the unit circle does number $\pi$ map to? +Back: $\langle -1, 0 \rangle$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers does the point $\langle -1, 0 \rangle$ on the unit circle map to? +Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers correspond to the highlighted point on the unit circle? +![[unit-circle-1-0.png]] +Back: $2 \pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers correspond to the highlighted point on the unit circle? +![[unit-circle-0-1.png]] +Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers correspond to the highlighted point on the unit circle? +![[unit-circle-n1-0.png]] +Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +Which real numbers correspond to the highlighted point on the unit circle? +![[unit-circle-0-n1.png]] +Back: $\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +*Why* does point $\langle 1, 0 \rangle$ on the unit circle coincide with real number $2\pi$? +Back: Because the circumference of the unit circle is $2\pi$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +*Why* does point $\langle -1, 0 \rangle$ on the unit circle coincide with real number $\pi$? +Back: Because half the circumference of the unit circle is $\pi$. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What is the "periodicity" of the unit circle? +Back: $2 \pi$ +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +%%ANKI +Basic +What property of the unit circle does its periodicity correspond to? +Back: Its circumference. +Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. + +END%% + +## Bibliography + +* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. \ No newline at end of file