3 % We have to load this before mjotex so that mjotex knows to define
4 % its glossary entries.
5 \usepackage[nonumberlist
]{glossaries
}
13 \begin{section
}{Algebra
}
14 If $R$ is a commutative ring, then $
\polyring{R
}{X,Y,Z
}$ is a
15 multivariate polynomial ring with indeterminates $X$, $Y$, and
16 $Z$, and coefficients in $R$. If $R$ is a moreover an integral
17 domain, then its fraction field is $
\Frac{R
}$.
20 \begin{section
}{Algorithm
}
21 An example of an algorithm (bogosort) environment.
24 \caption{Sort a list of numbers
}
26 \Require{A list of numbers $L$
}
27 \Ensure{A new, sorted copy $M$ of the list $L$
}
31 \While{$M$ is not sorted
}
32 \State{Rearrange $M$ randomly
}
40 \begin{section
}{Arrow
}
41 The identity operator on $V$ is $
\identity{V
}$. The composition of
42 $f$ and $g$ is $
\compose{f
}{g
}$. The inverse of $f$ is
43 $
\inverse{f
}$. If $f$ is a function and $A$ is a subset of its
44 domain, then the preimage under $f$ of $A$ is $
\preimage{f
}{A
}$.
47 \begin{section
}{Calculus
}
48 The gradient of $f :
\Rn \rightarrow \Rn[1]$ is $
\gradient{f
} :
52 \begin{section
}{Common
}
53 The function $f$ applied to $x$ is $f
\of{x
}$. We can group terms
54 like $a +
\qty{b - c
}$ or $a +
\qty{b -
\sqty{c - d
}}$. Here's a
55 set $
\set{1,
2,
3} =
\setc{n
\in \Nn[1]}{ n
\le 3 }$. Here's a pair
56 of things $
\pair{1}{2}$ or a triple of them $
\triple{1}{2}{3}$,
57 and the factorial of the number $
10$ is $
\factorial{10}$.
59 The Cartesian product of two sets $A$ and $B$ is
60 $
\cartprod{A
}{B
}$; if we take the product with $C$ as well, then
61 we obtain $
\cartprodthree{A
}{B
}{C
}$. The direct sum of $V$ and $W$
62 is $
\directsum{V
}{W
}$. Or three things,
63 $
\directsumthree{U
}{V
}{W
}$. How about more things? Like
64 $
\directsummany{k=
1}{\infty}{V_
{k
}} \ne
65 \cartprodmany{k=
1}{\infty}{V_
{k
}}$. Those direct sums and
66 cartesian products adapt nicely to display equations:
69 \directsummany{k=
1}{\infty}{V_
{k
}} \ne \cartprodmany{k=
1}{\infty}{V_
{k
}}.
71 Here are a few common tuple spaces that should not have a
72 superscript when that superscript would be one: $
\Nn[1]$,
73 $
\Zn[1]$, $
\Qn[1]$, $
\Rn[1]$, $
\Cn[1]$. However, if the
74 superscript is (say) two, then it appears: $
\Nn[2]$, $
\Zn[2]$,
75 $
\Qn[2]$, $
\Rn[2]$, $
\Cn[2]$.
77 We also have a few basic set operations, for example the union of
78 two or three sets: $
\union{A
}{B
}$, $
\unionthree{A
}{B
}{C
}$. And of
79 course with union comes intersection: $
\intersect{A
}{B
}$,
80 $
\intersectthree{A
}{B
}{C
}$. We can also take an arbitrary
81 (indexed) union and intersections of things, like
82 $
\unionmany{k=
1}{\infty}{A_
{k
}}$ or
83 $
\intersectmany{k=
1}{\infty}{B_
{k
}}$. The best part about those
84 is that they do the right thing in a display equation:
87 \unionmany{k=
1}{\infty}{A_
{k
}} =
\intersectmany{k=
1}{\infty}{B_
{k
}}
90 Finally, we have the four standard types of intervals in $
\Rn[1]$,
93 \intervaloo{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a < x < b
},\\
94 \intervaloc{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a < x
\le b
},\\
95 \intervalco{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a
\le x < b
},
\text{ and
}\\
96 \intervalcc{a
}{b
} &=
\setc{ x
\in \Rn[1]}{ a
\le x
\le b
}.
100 \begin{section
}{Complex
}
101 We sometimes want to conjugate complex numbers like
102 $
\compconj{a+bi
} = a - bi$.
105 \begin{section
}{Cone
}
106 The dual cone of $K$ is $
\dual{K
}$. Some familiar symmetric cones
107 are $
\Rnplus$, $
\Lnplus$, $
\Snplus$, and $
\Hnplus$. If cones
108 $K_
{1}$ and $K_
{2}$ are given, we can define $
\posops{K_
{1}}$,
109 $
\posops[K_
{2}]{K_
{1}}$, $
\Sof{K_
{1}}$, $
\Zof{K_
{1}}$,
110 $
\LL{K_
{1}}$, and $
\lyapunovrank{K_
{1}}$. We can also define $x
111 \gecone_{K
} y$, $x
\gtcone_{K
} y$, $x
\lecone_{K
} y$, and $x
112 \ltcone_{K
} y$ with respect to a cone $K$.
115 \begin{section
}{Convex
}
116 The conic hull of a set $X$ is $
\cone{X
}$; its affine hull is
117 $
\aff{X
}$, and its convex hull is $
\conv{X
}$. If $K$ is a cone,
118 then its lineality space is $
\linspace{K
}$, its lineality is
119 $
\lin{K
}$, and its extreme directions are $
\Ext{K
}$. The fact that
120 $F$ is a face of $K$ is denoted by $F
\faceof K$; if $F$ is a
121 proper face, then we write $F
\properfaceof K$.
124 \begin{section
}{Euclidean Jordan algebras
}
125 The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
129 \begin{section
}{Font
}
130 We can write things like Carathéodory and Güler and $
\mathbb{R
}$.
133 \begin{section
}{Linear algebra
}
134 The absolute value of $x$ is $
\abs{x
}$, or its norm is
135 $
\norm{x
}$. The inner product of $x$ and $y$ is $
\ip{x
}{y
}$ and
136 their tensor product is $
\tp{x
}{y
}$. The Kronecker product of
137 matrices $A$ and $B$ is $
\kp{A
}{B
}$. The adjoint of the operator
138 $L$ is $
\adjoint{L
}$, or if it's a matrix, then its transpose is
139 $
\transpose{L
}$. Its trace is $
\trace{L
}$. Another matrix-specific
140 concept is the Moore-Penrose pseudoinverse of $L$, denoted by
143 The span of a set $X$ is $
\spanof{X
}$, and its codimension is
144 $
\codim{X
}$. The projection of $X$ onto $V$ is $
\proj{V
}{X
}$. The
145 automorphism group of $X$ is $
\Aut{X
}$, and its Lie algebra is
146 $
\Lie{X
}$. We can write a column vector $x
\coloneqq
147 \colvec{x_
{1},x_
{2},x_
{3},x_
{4}}$ and turn it into a $
2 \times 2$
148 matrix with $
\matricize{x
}$. To recover the vector, we use
149 $
\vectorize{\matricize{x
}}$.
151 The set of all bounded linear operators from $V$ to $W$ is
152 $
\boundedops[W
]{V
}$. If $W = V$, then we write $
\boundedops{V
}$
155 The direct sum of $V$ and $W$ is $
\directsum{V
}{W
}$, of course,
156 but what if $W = V^
{\perp}$? Then we wish to indicate that fact by
157 writing $
\directsumperp{V
}{W
}$. That operator should survive a
158 display equation, too, and the weight of the circle should match
159 that of the usual direct sum operator.
162 Z =
\directsumperp{V
}{W
}\\
163 \oplus \oplusperp \oplus \oplusperp
166 Its form should also survive in different font sizes...
169 Z =
\directsumperp{V
}{W
}\\
170 \oplus \oplusperp \oplus \oplusperp
174 Z =
\directsumperp{V
}{W
}\\
175 \oplus \oplusperp \oplus \oplusperp
180 \begin{section
}{Listing
}
181 Here's an interactive SageMath prompt:
183 \begin{tcblisting
}{listing only,
186 listing options=
{language=sage,style=sage
}}
187 sage: K = Cone(
[ (
1,
0), (
0,
1)
])
188 sage: K.positive_operator_gens()
190 [1 0] [0 1] [0 0] [0 0]
191 [0 0],
[0 0],
[1 0],
[0 1]
195 However, the smart way to display a SageMath listing is to load it
196 from an external file (under the ``listings'' subdirectory):
198 \sagelisting{example
}
200 Keeping the listings in separate files makes it easy for the build
204 \begin{section
}{Miscellaneous
}
205 The cardinality of the set $X
\coloneqq \set{1,
2,
3}$ is $
\card{X
}
209 \begin{section
}{Proof by cases
}
212 There are two cases in the following proof.
215 The result should be self-evident once we have considered the
218 \begin{case
}[first case
]
219 Nothing happens in the first case.
221 \begin{case
}[second case
]
222 The same thing happens in the second case.
232 \renewcommand{\baselinestretch}{2}
234 Cases should display intelligently even when the
document is
241 \begin{case
}[first case
]
242 Nothing happens in the first case.
244 \begin{case
}[second case
]
245 The same thing happens in the second case.
252 \renewcommand{\baselinestretch}{1}
255 \begin{section
}{Theorems
}
289 \begin{section
}{Theorems (starred)
}
323 \begin{section
}{Topology
}
324 The interior of a set $X$ is $
\interior{X
}$. Its closure is
325 $
\closure{X
}$ and its boundary is $
\boundary{X
}$.
328 \setlength{\glslistdottedwidth}{.3\linewidth}
329 \setglossarystyle{listdotted
}
331 \printnoidxglossaries