From 7871ae52fec79ce4652a3990efdfadc09975378a Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Fri, 19 Apr 2024 20:44:51 -0600 Subject: [PATCH] Polynomials and delta-constants. --- .../plugins/obsidian-to-anki-plugin/data.json | 14 +- notes/_journal/2024-04-18.md | 11 - notes/_journal/2024-04-19.md | 15 ++ notes/_journal/2024-04/2024-04-18.md | 15 ++ notes/algebra/polynomials.md | 130 +++++++++++ notes/algebra/sequences/arithmetic.md | 8 + notes/algebra/sequences/delta-constant.md | 201 ++++++++++++++++++ notes/algebra/sequences/triangular-numbers.md | 8 + notes/combinatorics/combinations.md | 40 ---- 9 files changed, 387 insertions(+), 55 deletions(-) delete mode 100644 notes/_journal/2024-04-18.md create mode 100644 notes/_journal/2024-04-19.md create mode 100644 notes/_journal/2024-04/2024-04-18.md create mode 100644 notes/algebra/polynomials.md create mode 100644 notes/algebra/sequences/delta-constant.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 3b3a73d..91d9974 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -210,7 +210,7 @@ "_journal/2024-02-19.md": "30d16c5373deb9cb128d2e7934ae256a", "_journal/2024-02/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629", "combinatorics/permutations.md": "1b994b48798699655ee64df29c640251", - "combinatorics/combinations.md": "b83f325caa1a7a97a3873016d6ccefcc", + "combinatorics/combinations.md": "1109af89c5f3343ef978e4385c4b3819", "_journal/2024-02-20.md": "b85ba0eeeb16e30a602ccefabcc9763e", "_journal/2024-02/2024-02-19.md": "df1a9ab7ab89244021b3003c84640c78", "combinatorics/inclusion-exclusion.md": "c27b49ee03cc5ee854d0e8bd12a1d505", @@ -249,7 +249,7 @@ "algebra/sequences/index.md": "91ec81d3aa22d1baef2ab4b24736c43c", "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", - "algebra/sequences/triangular-numbers.md": "e833b69d767fb33594716cb0b2bd6aa3", + "algebra/sequences/triangular-numbers.md": "39a84ee317d3760a2eda7279c83e921a", "algebra/sequences/square-numbers.md": "171f7c5a8dac088afba40923ab86c68e", "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670", "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2", @@ -258,7 +258,7 @@ "_journal/2024-03-05.md": "e9a911c19bb4c0ff451db793248cb4bb", "_journal/2024-03/2024-03-04.md": "4948d90a08af2cff58c629c9a2e11ee4", "algebra/sequences/geometric.md": "57544cab59f0b8c28d4a11f0273a3119", - "algebra/sequences/arithmetic.md": "0e8d8168f04550cb506a218b298c1c7f", + "algebra/sequences/arithmetic.md": "6acb04ca2bbd2e39decd23a6451f2266", "_journal/2024-03-06.md": "ac7a3d764934f49b2be7aa76e402d853", "_journal/2024-03/2024-03-05.md": "94b28d0b9bc62cc0bd99d315fb7c6d30", "_journal/2024-03-07.md": "7bf68d6d81e89aa00f5ddd7510b69e3e", @@ -347,7 +347,13 @@ "_journal/2024-04-15.md": "c08bd7258def0c2a6b78039bb4aa03b9", "_journal/2024-04/2024-04-14.md": "037c77d0e11f2d58ffee61ea0a1708ab", "_journal/2024-04-16.md": "0bf6e2f2a3afab73d528cee88c4c1a92", - "_journal/2024-04/2024-04-15.md": "256253b0633d878ca58060162beb7587" + "_journal/2024-04/2024-04-15.md": "256253b0633d878ca58060162beb7587", + "algebra/polynomials.md": "6e20029b44fe0d0c4f35ef8ee4874d82", + "algebra/sequences/delta-constant.md": "f51bb01d5475004063e444e2acbcd0a3", + "_journal/2024-04-19.md": "a293087860a7f378507a96df0b09dd2b", + "_journal/2024-04/2024-04-18.md": "f6e5bee68dbef90a21ca92a846930a88", + "_journal/2024-04/2024-04-17.md": "331423470ea83fc990c1ee1d5bd3b3f1", + "_journal/2024-04/2024-04-16.md": "2f47dcbc01dd0995e59ae1759122b416" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-04-18.md b/notes/_journal/2024-04-18.md deleted file mode 100644 index f86bf9b..0000000 --- a/notes/_journal/2024-04-18.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -title: "2024-04-18" ---- - -- [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/_journal/2024-04-19.md b/notes/_journal/2024-04-19.md new file mode 100644 index 0000000..1825184 --- /dev/null +++ b/notes/_journal/2024-04-19.md @@ -0,0 +1,15 @@ +--- +title: "2024-04-19" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [x] Interview Prep (1 Practice Problem) +- [x] Log Work Hours (Max 3 hours) + +* Hide-and-seek application + * Revisions to settings. Primarily flattening the structure. +* Read 2.3 "Polynomial Fitting" of "Discrete Mathematics: An Open Introduction". \ No newline at end of file diff --git a/notes/_journal/2024-04/2024-04-18.md b/notes/_journal/2024-04/2024-04-18.md new file mode 100644 index 0000000..7bac44f --- /dev/null +++ b/notes/_journal/2024-04/2024-04-18.md @@ -0,0 +1,15 @@ +--- +title: "2024-04-18" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [x] Interview Prep (1 Practice Problem) +- [x] Log Work Hours (Max 3 hours) + +* Discussed first chapter of "Grokking the System Design Interview" as well as chapter 6 of "Designing Data-Intensive Applications" with Mike. +* Applied to several more job openings. +* Revised presence tracking and host updates in hide-and-seek application. \ No newline at end of file diff --git a/notes/algebra/polynomials.md b/notes/algebra/polynomials.md new file mode 100644 index 0000000..cbfe364 --- /dev/null +++ b/notes/algebra/polynomials.md @@ -0,0 +1,130 @@ +--- +title: Polynomials +TARGET DECK: Obsidian::STEM +FILE TAGS: algebra::polynomial +tags: + - algebra + - polynomial +--- + +## Overview + +Given nonnegative integer $d$, a **polynomial in $n$ of degree $d$** is a function $p(n)$ of the form $$p(n) = \sum_{i=0}^d a_i n^i$$ +The coefficients of $p(n)$ are $a_0, a_1, \ldots, a_d$. Furthermore, $a_d \neq 0$. + +%%ANKI +Basic +Using sigma notation, a polynomial in $n$ of degree $d$ is a function of what form? +Back: $p(n) = \sum_{i=0}^d a_in^i$ where $a_d \neq 0$. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What four algebraic operations are permitted in a polynomial? +Back: Addition, subtraction, multiplication, and exponentiation. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What is $d$ in "a polynomial in $n$ of degree $d$"? +Back: $d$ is a nonnegative integer. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What is $n$ in "a polynomial in $n$ of degree $d$"? +Back: The polynomial's indeterminate. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +Which coefficient is special in a polynomial and why? +Back: That attached to the monomial with highest degree because it cannot be zero. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What are the coefficients of a polynomial? +Back: The constants of the monomials found in the polynomial. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What name is given to a degree-0 polynomial? +Back: A constant. +Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). + +END%% + +%%ANKI +Basic +What name is given to a degree-1 polynomial? +Back: A monomial. +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 name is given to a degree-2 polynomial? +Back: A binomial. +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 binomial? +Back: A polynomial containing two terms. +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 polynomial fitting? +Back: The solving of a linear system to find the coefficients of a polynomial. +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 +Polynomial fitting is a strategy for discovering what part of a polynomial? +Back: The coefficients. +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 degree of a polynomial? +Back: The highest degree of the monomials with non-zero coefficients. +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 many data points are required to fit a polynomial? +Back: $k + 1$ where $k$ is the degree of the polynomial. +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%% + +## 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). +* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). \ No newline at end of file diff --git a/notes/algebra/sequences/arithmetic.md b/notes/algebra/sequences/arithmetic.md index 86a4a5a..21ef2ec 100644 --- a/notes/algebra/sequences/arithmetic.md +++ b/notes/algebra/sequences/arithmetic.md @@ -150,6 +150,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% +%%ANKI +Basic +Why is a sequence of partial sums named the way it is? +Back: Each term is found by adding a finite number of infinite terms. +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%% + ## 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). \ No newline at end of file diff --git a/notes/algebra/sequences/delta-constant.md b/notes/algebra/sequences/delta-constant.md new file mode 100644 index 0000000..d6064df --- /dev/null +++ b/notes/algebra/sequences/delta-constant.md @@ -0,0 +1,201 @@ +--- +title: ∆ᵏ-constant Sequence +TARGET DECK: Obsidian::STEM +FILE TAGS: algebra::sequence algebra::polynomial +tags: + - algebra + - sequence +--- + +## Overview + +Let $(a_n)$ be a sequence. We define the **$k$th differences of $(a_n)$** recursively: + +* The $0$th differences of $(a_n)$ is $(a_n)$. +* The $k$th differences of $(a_n)$ is the sequence given by subtracting consecutive terms of the $(k-1)st$ differences of $(a_n)$. + +A sequence is said to be **$\Delta^k$-constant** if the $k$th differences are constant. + +> The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $\Delta^k$-constant. + +This is the discrete analogue to (continuous) derivatives of polynomials. + +%%ANKI +Basic +What are the $0$th differences of $(a_n)_{n \geq 0}$? +Back: $(a_n)_{n \geq 0}$ +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 are the $1$st differences of $(a_n)_{n \geq 0}$? +Back: $(b_n)_{n \geq 1}$ where $b_n = a_n - a_{n - 1}$. +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 +How do we refer to the $1$st differences of $(a_n)$ more naturally? +Back: As "the differences of $(a_n)$." +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 is the base case of the recursive definition of the $k$th differences of $(a_n)$? +Back: $k = 0$ +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 recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$? +Back: The $(k + 1)$st differences is the differences of the $k$th differences. +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 does it mean for $(a_n)$ to be $\Delta^k$-constant? +Back: The $k$th differences of $(a_n)$ is constant. +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 can be said about the closed formula of a $\Delta^k$-constant sequence? +Back: It is a polynomial with degree $k$. +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 +We say a sequence is {$\Delta^k$-constant} when the {$k$th differences is constant}. +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 +$\Delta^k$-constant sequences are a discrete analogue to what calculus concept? +Back: Derivatives. +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). +Tags: calculus + +END%% + +%%ANKI +Basic +What kind of mathematical expression do $\Delta^k$-constant sequences relate to? +Back: Polynomials. +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). +Tags: calculus + +END%% + +%%ANKI +Basic +How can we prove every quadratic sequence, say $(a_n)$, has arithmetic differences? +Back: By showing $a_{n+1} - a_n$ is linear. +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 can we prove every cubic sequence, say $(a_n)$, has quadratic differences? +Back: By showing $a_{n+1} - a_n$ is quadratic. +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 +The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences {is constant}. +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 are arithmetic sequences defined in terms of $\Delta^k$ polynomials? +Back: A sequence is arithmetic if and only if it is $\Delta^1$-constant. +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 are geometric sequences defined in terms of $\Delta^k$ polynomials? +Back: N/A +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 +*Why* can't a geometric sequence be $\Delta^k$-constant for some $k \geq 0$? +Back: Because the closed formula of a geometric sequence is not a polynomial. +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 +Suppose $(a_n)$ is a $\Delta^2$-constant sequence. What general form describes its closed formula? +Back: $an^2 + bn + c$ +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 +Suppose $(a_n)$ is a $\Delta^3$-constant sequence. What general form describes its closed formula? +Back: $an^3 + bn^2 + cn + d$ +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 +Suppose $(a_n)$ is a $\Delta^k$-constant sequence. Where is $k$ repeated in $(a_n)$'s closed formula? +Back: At the largest degree of the polynomial. +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 are the terms in a $\Delta^0$-constant sequences? +Back: $c, c, c, \ldots$ for some constant $c$. +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 $\Delta^k$-constant sequences relate to polynomial fitting? +Back: The closed formula of a $\Delta^k$-constant sequence is a polynomial we can fit. +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 many data points of a $\Delta^k$-constant sequence are needed to polynomial fit its closed formula? +Back: $k + 1$ +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%% + +## 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). \ No newline at end of file diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md index 1e31fdd..1b80fbf 100644 --- a/notes/algebra/sequences/triangular-numbers.md +++ b/notes/algebra/sequences/triangular-numbers.md @@ -254,6 +254,14 @@ Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://e END%% +%%ANKI +Basic +The triangular numbers is the sequence of partial sums of what sequence? +Back: $a_n = n$ +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%% + ## 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). diff --git a/notes/combinatorics/combinations.md b/notes/combinatorics/combinations.md index 9e5e4bc..6054eaf 100644 --- a/notes/combinatorics/combinations.md +++ b/notes/combinatorics/combinations.md @@ -525,46 +525,6 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% -%%ANKI -Basic -What is a degree-0 polynomial? -Back: A constant. -Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). - -END%% - -%%ANKI -Basic -What name is given to a degree-0 polynomial? -Back: A constant. -Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). - -END%% - -%%ANKI -Basic -What name is given to a degree-1 polynomial? -Back: A monomial. -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 name is given to a degree-2 polynomial? -Back: A binomial. -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 binomial? -Back: A polynomial containing two terms. -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 Why are binomial coefficients named the way they are?