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Mark
2025-09-24 22:02:23 -07:00
parent 69d835a2d2
commit 0b7acaf5ae
8 changed files with 31 additions and 54 deletions
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49
src/Advanced/Tropical Polynomials/parts/00 arithmetic.typ
View File

@ -26,9 +26,9 @@ $
#solution([
- Is tropical addition commutative?\
Yes, $min(min(x,y),z) = min(x,y,z) = min(x,min(y,z))$
Yes, $min(min(x, y), z) = min(x, y, z) = min(x, min(y, z))$
- Is tropical addition associative? \
Yes, $min(x,y) = min(y,x)$
Yes, $min(x, y) = min(y, x)$
- Is there a tropical additive identity? \
No. There is no $n$ where $x <= n$ for all real $x$
])
@ -117,7 +117,7 @@ Do tropical multiplicative inverses always exist? \
Is tropical multiplication distributive over addition? \
#note([Does $x #tm (y #tp z) = x #tm y #tp x #tm z$?])
#solution([Yes, $x + min(y,z) = min(x+y, x+z)$])
#solution([Yes, $x + min(y, z) = min(x+y, x+z)$])
#v(1fr)
@ -134,14 +134,7 @@ Fill the following tropical addition and multiplication tables
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tp$],
[$1$],
[$2$],
[$3$],
[$4$],
[$#sym.infinity$],
),
table.header([$#tp$], [$1$], [$2$], [$3$], [$4$], [$#sym.infinity$]),
box(inset: 3pt, $1$), [], [], [], [], [],
box(inset: 3pt, $2$), [], [], [], [], [],
@ -152,14 +145,7 @@ Fill the following tropical addition and multiplication tables
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tm$],
[$0$],
[$1$],
[$2$],
[$3$],
[$4$],
),
table.header([$#tm$], [$0$], [$1$], [$2$], [$3$], [$4$]),
box(inset: 3pt, $0$), [], [], [], [], [],
box(inset: 3pt, $1$), [], [], [], [], [],
@ -178,14 +164,7 @@ Fill the following tropical addition and multiplication tables
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tp$],
[$1$],
[$2$],
[$3$],
[$4$],
[$#sym.infinity$],
),
table.header([$#tp$], [$1$], [$2$], [$3$], [$4$], [$#sym.infinity$]),
box(inset: 3pt, $1$),
box(inset: 3pt, $1$),
@ -225,14 +204,7 @@ Fill the following tropical addition and multiplication tables
table(
columns: (col, col, col, col, col, col),
align: center,
table.header(
[$#tm$],
[$0$],
[$1$],
[$2$],
[$3$],
[$4$],
),
table.header([$#tm$], [$0$], [$1$], [$2$], [$3$], [$4$]),
box(inset: 3pt, $0$),
box(inset: 3pt, $0$),
@ -281,10 +253,9 @@ Adjacent parenthesis imply tropical multiplication
#solution([
$
(x #tp 2)(x #tp 3)
&= x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
&= x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
&= x^2 #tp 2x #tp 5
(x #tp 2)(x #tp 3) & = x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
& = x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
& = x^2 #tp 2x #tp 5
$
Also, $f(1) = 2$ and $f(4) = 5$.

2
src/Advanced/Tropical Polynomials/parts/02 cubic.typ
View File

@ -166,7 +166,7 @@ Find a formula for each $C_i$ in terms of $c_0, c_1, ..., c_n$.
#solution([
$
A_j & = min_(l<=j<k)( (a_l - a_k) / (k-l) (k-j) + a_k ) \
A_j & = min_(l<=j<k)( (a_l - a_k) / (k-l) (k-j) + a_k ) \
& = min_(l<=j<k)( a_l (k-j) / (k-l) + a_k (j-l) / (k-l) )
$