Polynomials and delta-constants.

c-declarations
Joshua Potter 2024-04-19 20:44:51 -06:00
parent 5f51ad65f1
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"Basic": [ "Basic": [

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---
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)

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---
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".

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---
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.

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---
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).
<!--ID: 1713580808758-->
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).
<!--ID: 1713580808763-->
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).
<!--ID: 1713580808766-->
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).
<!--ID: 1713580808769-->
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).
<!--ID: 1713580808772-->
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).
<!--ID: 1713580808776-->
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).
<!--ID: 1708974221749-->
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).
<!--ID: 1708974221752-->
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).
<!--ID: 1708974221755-->
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).
<!--ID: 1708368078759-->
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).
<!--ID: 1713580109018-->
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).
<!--ID: 1713580808780-->
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).
<!--ID: 1713580109082-->
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).
<!--ID: 1713580109089-->
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).

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@ -150,6 +150,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
<!--ID: 1709666305449--> <!--ID: 1709666305449-->
END%% 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).
<!--ID: 1713580109297-->
END%%
## Bibliography ## 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). * 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).

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---
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).
<!--ID: 1713580109096-->
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).
<!--ID: 1713580109118-->
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).
<!--ID: 1713580109125-->
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).
<!--ID: 1713580109133-->
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).
<!--ID: 1713580109153-->
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).
<!--ID: 1713580109175-->
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).
<!--ID: 1713580109182-->
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).
<!--ID: 1713580109189-->
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
<!--ID: 1713580109197-->
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
<!--ID: 1713580109203-->
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).
<!--ID: 1713580109225-->
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).
<!--ID: 1713580109232-->
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).
<!--ID: 1713580109237-->
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).
<!--ID: 1713580109244-->
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).
<!--ID: 1713580109250-->
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).
<!--ID: 1713580109255-->
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).
<!--ID: 1713580109261-->
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).
<!--ID: 1713580109265-->
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).
<!--ID: 1713580109270-->
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).
<!--ID: 1713580109274-->
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).
<!--ID: 1713580109280-->
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).
<!--ID: 1713580109285-->
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).

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@ -254,6 +254,14 @@ Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://e
<!--ID: 1709419325981--> <!--ID: 1709419325981-->
END%% 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).
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## Bibliography ## 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). * 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).

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%%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).
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%%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).
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%%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).
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%%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).
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%%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).
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%%ANKI %%ANKI
Basic Basic
Why are binomial coefficients named the way they are? Why are binomial coefficients named the way they are?