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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>
2025-01-22 13:38:51 -08:00
313c4fb4c6 Added "Tropical Polynomials" handout
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2025-01-22 12:40:34 -08:00
c4ea1aa5c5 Scripts and CI 2025-01-22 12:40:29 -08:00
5b84604d79 Warm-ups 2025-01-22 12:31:08 -08:00
03a2e920f8 Intermediate handouts 2025-01-22 12:31:06 -08:00
0ae3cdfb22 Advanced handouts 2025-01-22 12:30:56 -08:00
13 changed files with 19 additions and 38 deletions

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@ -1,5 +1,4 @@
{
"latex-workshop.latex.recipe.default": "latexmk (xelatex)",
"tinymist.formatterPrintWidth": 80,
"tinymist.typstExtraArgs": ["--package-path=./lib/typst"]
"tinymist.formatterPrintWidth": 80
}

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@ -1,3 +0,0 @@
[authors."mark"]
email = "mark@betalupi.com"
webpage = "betalupi.com"

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@ -1,6 +0,0 @@
[package]
name = "handout"
version = "0.1.0"
entrypoint = "handout.typ"
authors = []
license = "GPL"

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@ -1,5 +1,3 @@
/// Typst handout library, used for all documents in this repository.
/// If false, hide instructor info.
///
@ -240,8 +238,8 @@
set page(
margin: 20mm,
width: 8in,
height: 11.5in,
width: 8.5in,
height: 11in,
footer: align(
center,
context counter(page).display(),

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@ -1,4 +1,4 @@
#import "@local/handout:0.1.0": *
#import "./handout.typ": *
#import "@preview/cetz:0.3.1"

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@ -1,4 +1,4 @@
#import "@local/handout:0.1.0": *
#import "./handout.typ": *
#show: doc => handout(
doc,

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@ -1,4 +1,4 @@
#import "@local/handout:0.1.0": *
#import "../handout.typ": *
#import "../macros.typ": *
= Tropical Arithmetic
@ -62,7 +62,8 @@ Let's expand $#sym.RR$ to include a tropical additive identity.
#problem()
Do tropical additive inverses exist? \
#note([
Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$?
Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$? \
Remember that $#sym.infinity$ is the additive identity.
])
#solution([
@ -277,7 +278,7 @@ Fill the following tropical addition and multiplication tables
#problem()
Expand and simplify $f(x) = (x #tp 2)(x #tp 3)$, then evaluate $f(1)$ and $f(4)$ \
Adjacent parenthesis imply tropical multiplication
#hint([Adjacent parenthesis imply tropical multiplication])
#solution([
$

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@ -1,4 +1,4 @@
#import "@local/handout:0.1.0": *
#import "../handout.typ": *
#import "../macros.typ": *
#import "@preview/cetz:0.3.1"
@ -20,7 +20,7 @@ for some nonnegative integer $n$ and coefficients $c_0, c_1, ..., c_n$. \
The _degree_ of a polynomial is the largest $n$ for which $c_n$ is nonzero.
#theorem()
The _fundamental theorem of algebra_ implies that any non-constant polynomial with real coefficients
The _fundamental theorem of algebra_ states that any non-constant polynomial with real coefficients
can be written as a product of polynomials of degree 1 or 2 with real coefficients.
#v(2mm)
@ -30,8 +30,8 @@ can be written as $(x^2 + 2x+5)(x-2)(x+4)(x+4)$
#v(2mm)
A similar theorem exists for polynomials with complex coefficients. \
These coefficients may be found using the roots of this polynomial. \
As it turns out, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on approximate roots found by computers.
These coefficients may be found using the _roots_ of this polynomial. \
As you already know, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on appropriate roots found by computers.
#v(2mm)
In this section, we will analyze tropical polynomials:
@ -100,7 +100,7 @@ In other words, find $r$ and $s$ so that
),
)
we will call $r$ and $s$ the _roots_ of $f$.
#note([Naturally, we will call $r$ and $s$ the _roots_ of $f$.])
#solution([
Because $(x #tp r)(x #tp s) = x^2 #tp (r #tp s)x #tp s r$, we must have $r #tp s = 1$ and $r #tm s = 4$. \
@ -116,7 +116,8 @@ we will call $r$ and $s$ the _roots_ of $f$.
#v(1fr)
#problem()
How can we use the graph to determine these roots?
Can you see the roots of this polynomial in the graph? \
#hint([Yes, you can. What "features" do the roots correspond to?])
#solution([The roots are the corners of the graph.])
@ -316,7 +317,7 @@ into linear factors.
#v(2mm)
Whenever we say "the roots of $f$", we really mean "the roots of $accent(f, macron)$." \
$f$ and $accent(f, macron)$ might be the same polynomial.
Also, $f$ and $accent(f, macron)$ might be the same polynomial.
#problem()
If $f(x) = a x^2 #tp b x #tp c$, then $accent(f, macron)(x) = a x^2 #tp B x #tp c$ for some $B$. \

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@ -1,4 +1,4 @@
#import "@local/handout:0.1.0": *
#import "../handout.typ": *
#import "../macros.typ": *
#import "@preview/cetz:0.3.1"

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@ -4,8 +4,3 @@ title = "Odd Dice"
[publish]
handout = true
solutions = true
[[attribution]]
who = "mark"
when = 2024-02-13
what = "Initial version of handout"

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@ -143,12 +143,10 @@ def build_typst(source_dir: Path, out_subdir: Path) -> IndexEntry | None:
[
TYPST_PATH,
"compile",
"--package-path",
f"{ROOT}/lib/typst",
"--ignore-system-fonts",
"main.typ",
"--input",
"show_solutions=false",
"main.typ",
f"{out}/{handout_file}",
],
cwd=source_dir,
@ -166,8 +164,6 @@ def build_typst(source_dir: Path, out_subdir: Path) -> IndexEntry | None:
[
TYPST_PATH,
"compile",
"--package-path",
f"{ROOT}/lib/typst",
"--ignore-system-fonts",
"main.typ",
f"{out}/{solutions_file}",