3 % Setting hypertexnames=false forces hyperref to use a consistent
4 % internal counter for proposition/equation references rather than
5 % being clever, which doesn't work after we reset those counters.
6 \usepackage[hypertexnames=false
]{hyperref
}
13 % We have to load this after hyperref, so that links work, but before
14 % mjotex so that mjotex knows to define its glossary entries.
15 \usepackage[nonumberlist
]{glossaries
}
16 \makenoidxglossaries{}
18 % If you want an index, we can do that too. You'll need to define
19 % the "INDICES" variable in the GNUmakefile, though.
24 \usepackage{mathtools
}
28 \begin{section
}{Algebra
}
29 If $R$ is a commutative ring
\index{commutative ring
}, then
30 $
\polyring{R
}{X,Y,Z
}$ is a multivariate polynomial ring with
31 indeterminates $X$, $Y$, and $Z$, and coefficients in $R$. If $R$
32 is a moreover an integral domain, then its fraction field is
33 $
\Frac{R
}$. If $x,y,z
\in R$, then $
\ideal{\set{x,y,z
}}$ is the
34 ideal generated by $
\set{x,y,z
}$, which is defined to be the
35 smallest ideal in $R$ containing that set. Likewise, if we are in
36 an algebra $
\mathcal{A
}$ and if $x,y,z
\in \mathcal{A
}$, then
37 $
\alg{\set{x,y,z
}}$ is the smallest subalgebra of $
\mathcal{A
}$
38 containing the set $
\set{x,y,z
}$.
40 If $R$ has a multiplicative identity (that is, a unit) element,
41 then that element is denoted by $
\unit{R
}$.
44 \begin{section
}{Algorithm
}
45 An example of an algorithm (bogosort) environment.
48 \caption{Sort a list of numbers
}
50 \Require{A list of numbers $L$
}
51 \Ensure{A new, sorted copy $M$ of the list $L$
}
55 \While{$M$ is not sorted
}
56 \State{Rearrange $M$ randomly
}
64 \begin{section
}{Arrow
}
65 The constant function that always returns $a$ is $
\const{a
}$. The
66 identity operator on $V$ is $
\identity{V
}$. The composition of $f$
67 and $g$ is $
\compose{f
}{g
}$. The inverse of $f$ is
68 $
\inverse{f
}$. If $f$ is a function and $A$ is a subset of its
69 domain, then the preimage under $f$ of $A$ is $
\preimage{f
}{A
}$.
72 \begin{section
}{Calculus
}
73 The gradient of $f :
\Rn \rightarrow \Rn[1]$ is $
\gradient{f
} :
77 \begin{section
}{Common
}
78 The function $f$ applied to $x$ is $f
\of{x
}$, and the restriction
79 of $f$ to a subset $X$ of its domain is $
\restrict{f
}{X
}$. We can
80 group terms like $a +
\qty{b - c
}$ or $a +
\qty{b -
\sqty{c -
81 d
}}$. The tuples go up to seven, for now:
88 Triple: $
\triple{1}{2}{3}$,
91 Quadruple: $
\quadruple{1}{2}{3}{4}$,
94 Qintuple: $
\quintuple{1}{2}{3}{4}{5}$,
97 Sextuple: $
\sextuple{1}{2}{3}{4}{5}{6}$,
100 Septuple: $
\septuple{1}{2}{3}{4}{5}{6}{7}$.
104 The factorial of the number $
10$ is $
\factorial{10}$, and the
105 least common multiple of $
4$ and $
6$ is $
\lcm{\set{4,
6}} =
108 The direct sum of $V$ and $W$ is $
\directsum{V
}{W
}$. Or three
109 things, $
\directsumthree{U
}{V
}{W
}$. How about more things? Like
110 $
\directsummany{k=
1}{\infty}{V_
{k
}}$. Those direct sums
111 adapt nicely to display equations:
114 \directsummany{k=
1}{\infty}{V_
{k
}} \ne \emptyset.
117 Here are a few common tuple spaces that should not have a
118 superscript when that superscript would be one: $
\Nn[1]$,
119 $
\Zn[1]$, $
\Qn[1]$, $
\Rn[1]$, $
\Cn[1]$. However, if the
120 superscript is (say) two, then it appears: $
\Nn[2]$, $
\Zn[2]$,
121 $
\Qn[2]$, $
\Rn[2]$, $
\Cn[2]$. The symbols $
\Fn[1]$, $
\Fn[2]$,
122 et cetera, are available for use with a generic field.
124 Finally, we have the four standard types of intervals in $
\Rn[1]$,
127 \intervaloo{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a < x < b
},\\
128 \intervaloc{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a < x
\le b
},\\
129 \intervalco{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a
\le x < b
},
\text{ and
}\\
130 \intervalcc{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a
\le x
\le b
}.
134 \begin{section
}{Complex
}
135 We sometimes want to conjugate complex numbers like
136 $
\compconj{a+bi
} = a - bi$.
139 \begin{section
}{Cone
}
140 The dual cone of $K$ is $
\dual{K
}$. Some familiar symmetric cones
141 are $
\Rnplus$, $
\Lnplus$, $
\Snplus$, and $
\Hnplus$. If cones
142 $K_
{1}$ and $K_
{2}$ are given, we can define $
\posops{K_
{1}}$,
143 $
\posops[K_
{2}]{K_
{1}}$, $
\Sof{K_
{1}}$, $
\Zof{K_
{1}}$,
144 $
\LL{K_
{1}}$, and $
\lyapunovrank{K_
{1}}$. We can also define $x
145 \gecone_{K
} y$, $x
\gtcone_{K
} y$, $x
\lecone_{K
} y$, and $x
146 \ltcone_{K
} y$ with respect to a cone $K$.
149 \begin{section
}{Convex
}
150 The conic hull of a set $X$ is $
\cone{X
}$; its affine hull is
151 $
\aff{X
}$, and its convex hull is $
\conv{X
}$. If $K$ is a cone,
152 then its lineality space is $
\linspace{K
}$, its lineality is
153 $
\lin{K
}$, and its extreme directions are $
\Ext{K
}$. The fact that
154 $F$ is a face of $K$ is denoted by $F
\faceof K$; if $F$ is a
155 proper face, then we write $F
\properfaceof K$.
158 \begin{section
}{Euclidean Jordan algebras
}
159 The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
163 \begin{section
}{Font
}
164 We can write things like Carathéodory and Güler and
165 $
\mathbb{R
}$. The PostScript Zapf Chancery font is also available
166 in both upper- and lower-case:
169 \begin{item
}$
\mathpzc{abcdefghijklmnopqrstuvwxyz
}$
\end{item
}
170 \begin{item
}$
\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ
}$
\end{item
}
174 \begin{section
}{Linear algebra
}
175 The absolute value of $x$ is $
\abs{x
}$, or its norm is
176 $
\norm{x
}$. The inner product of $x$ and $y$ is $
\ip{x
}{y
}$ and
177 their tensor product is $
\tp{x
}{y
}$. The Kronecker product of
178 matrices $A$ and $B$ is $
\kp{A
}{B
}$. The adjoint of the operator
179 $L$ is $
\adjoint{L
}$, or if it's a matrix, then its transpose is
180 $
\transpose{L
}$. Its trace is $
\trace{L
}$, and its spectrum---the
181 set of its eigenvalues---is $
\spectrum{L
}$. Another
182 matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
183 denoted by $
\pseudoinverse{L
}$. Finally, the rank of a matrix $L$
184 is $
\rank{L
}$. As far as matrix spaces go, we have the $n$-by-$n$
185 real-symmetric and complex-Hermitian matrices $
\Sn$ and $
\Hn$
186 respectively; however $
\Sn[1]$ and $
\Hn[1]$ do not automatically
187 simplify because the ``$n$'' does not indicate the arity of a
188 Cartesian product in this case. A handy way to represent the
189 matrix $A
\in \Rn[n
\times n
]$ whose only non-zero entries are on
190 the diagonal is $
\diag{\colvec{A_
{11},A_
{22},
\ldots,A_
{nn
}}}$.
192 The span of a set $X$ is $
\spanof{X
}$, and its codimension is
193 $
\codim{X
}$. The projection of $X$ onto $V$ is $
\proj{V
}{X
}$. The
194 automorphism group of $X$ is $
\Aut{X
}$, and its Lie algebra is
195 $
\Lie{X
}$. We can write a column vector $x
\coloneqq
196 \colvec{x_
{1},x_
{2},x_
{3},x_
{4}}$ and turn it into a $
2 \times 2$
197 matrix with $
\matricize{x
}$. To recover the vector, we use
198 $
\vectorize{\matricize{x
}}$.
200 The set of all bounded linear operators from $V$ to $W$ is
201 $
\boundedops[W
]{V
}$. If $W = V$, then we write $
\boundedops{V
}$
204 If you want to solve a system of equations, try Cramer's
205 rule~
\cite{ehrenborg
}.
207 The direct sum of $V$ and $W$ is $
\directsum{V
}{W
}$, of course,
208 but what if $W = V^
{\perp}$? Then we wish to indicate that fact by
209 writing $
\directsumperp{V
}{W
}$. That operator should survive a
210 display equation, too, and the weight of the circle should match
211 that of the usual direct sum operator.
214 Z =
\directsumperp{V
}{W
}\\
215 \oplus \oplusperp \oplus \oplusperp
218 Its form should also survive in different font sizes
\ldots
221 Z =
\directsumperp{V
}{W
}\\
222 \oplus \oplusperp \oplus \oplusperp
226 Z =
\directsumperp{V
}{W
}\\
227 \oplus \oplusperp \oplus \oplusperp
232 \begin{section
}{Listing
}
233 Here's an interactive SageMath prompt:
235 \begin{tcblisting
}{listing only,
238 listing options=
{language=sage,style=sage
}}
239 sage: K = Cone(
[ (
1,
0), (
0,
1)
])
240 sage: K.positive_operator_gens()
242 [1 0] [0 1] [0 0] [0 0]
243 [0 0],
[0 0],
[1 0],
[0 1]
247 However, the smart way to display a SageMath listing is to load it
248 from an external file (under the ``listings'' subdirectory):
250 \sagelisting{example
}
252 Keeping the listings in separate files makes it easy for the build
256 \begin{section
}{Proof by cases
}
259 There are two cases in the following proof.
262 The result should be self-evident once we have considered the
265 \begin{case
}[first case
]
266 Nothing happens in the first case.
268 \begin{case
}[second case
]
269 The same thing happens in the second case.
279 \renewcommand{\baselinestretch}{2}
281 Cases should display intelligently even when the
document is
288 \begin{case
}[first case
]
289 Nothing happens in the first case.
291 \begin{case
}[second case
]
292 The same thing happens in the second case.
299 \renewcommand{\baselinestretch}{1}
302 \begin{section
}{Set theory
}
303 Here's a set $
\set{1,
2,
3} =
\setc{n
\in \Nn[1]}{ n
\le 3 }$. The
304 cardinality of the set $X
\coloneqq \set{1,
2,
3}$ is $
\card{X
} =
305 3$, and its powerset is $
\powerset{X
}$.
307 We also have a few basic set operations, for example the union of
308 two or three sets: $
\union{A
}{B
}$, $
\unionthree{A
}{B
}{C
}$. And of
309 course with union comes intersection: $
\intersect{A
}{B
}$,
310 $
\intersectthree{A
}{B
}{C
}$. The Cartesian product of two sets $A$
311 and $B$ is there too: $
\cartprod{A
}{B
}$. If we take the product
312 with $C$ as well, then we obtain $
\cartprodthree{A
}{B
}{C
}$.
314 We can also take an arbitrary (indexed) union, intersection, or
315 Cartesian product of things, like
316 $
\unionmany{k=
1}{\infty}{A_
{k
}}$,
317 $
\intersectmany{k=
1}{\infty}{B_
{k
}}$, or
318 $
\cartprodmany{k=
1}{\infty}{C_
{k
}}$. The best part about those is
319 that they do the right thing in a display equation:
322 \unionmany{k=
1}{\infty}{A_
{k
}}
324 \intersectmany{k=
1}{\infty}{B_
{k
}}
326 \cartprodmany{k=
1}{\infty}{C_
{k
}}.
331 \begin{section
}{Theorems
}
365 \begin{section
}{Theorems (starred)
}
399 \begin{section
}{Topology
}
400 The interior of a set $X$ is $
\interior{X
}$. Its closure is
401 $
\closure{X
}$ and its boundary is $
\boundary{X
}$.
404 \setlength{\glslistdottedwidth}{.3\linewidth}
405 \setglossarystyle{listdotted
}
407 \printnoidxglossaries{}
409 \bibliographystyle{mjo
}
410 \bibliography{local-references
}