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Wzory – katalog
Dział Wydawnictw IMPAN
[email protected]
• Uwaga ogólna: wymagany styl amsart lub \usepackage{amsmath}
1
(1.1)
(1.2)
Kilka wzorów – układy warunków
aaaaaaaaaa = b,
cc = xxx,
mmmmmmmmmmmmm = 0
dd = yyy,
for all i = 1, . . . , n.
\begin{gather}
aaaaaaaaaa = b, \quad\
cc = xxx,
\quad\
dd = yyy, \label{E:g1}\\
mmmmmmmmmmmmm = 0
\quad\
\text{for all }i=1,\ldots,n. \label{E:g2}
\end{gather}
aaaaaaaaaa = b,
(1.3)
cc = xxx,
mmmmmmmmmmmmm = 0
dd = yyy,
for all i = 1, . . . , n.
\begin{gather}
aaaaaaaaaa = b, \quad\
cc = xxx,
\quad\
dd = yyy, \notag\\
mmmmmmmmmmmmm = 0
\quad\
\text{for all }i=1,\ldots,n. \label{E:g3}
\end{gather}
aaaaaaaaaa = b,
cc = xxx,
mmmmmmmmmmmmm = 0
dd = yyy,
for all i = 1, . . . , n.
\begin{gather*}
aaaaaaaaaa = b, \quad\
cc = xxx,
\quad\
dd = yyy,\\
mmmmmmmmmmmmm = 0
\quad\
\text{for all }i=1,\ldots,n.
\end{gather*}
1
2
(1.4)
Wzory – katalog
aaaaaaaaaa = b,
cc = xxx,
mmmmmmmmmmmmm = 0
dd = yyy,
for all i = 1, . . . , n.
\begin{equation} \label{E:g4}
\begin{gathered}
aaaaaaaaaa = b, \quad\
cc = xxx,
\quad\
dd = yyy,\\
mmmmmmmmmmmmm = 0
\quad\
\text{for all }i=1,\ldots,n.
\end{gathered}
\end{equation}
(1.5)
xxxxx = yyyyyyyyyyyyyy
(1.6)
+ zzzzzzzzzzzzzzzzzz,
bbb = tttttttttttttttttt,
(1.7)
hh = vvvvvvvvvvv.
\begin{align}
xxxxx &= yyyyyyyyyyyyyy
\label{E:a1}\\
&\quad + zzzzzzzzzzzzzzzzzz, \notag\\
bbb &= tttttttttttttttttt, \label{E:a2}\\
hh &= vvvvvvvvvvv.
\label{E:a3}
\end{align}
xxxxx = yyyyyyyyyyyyyy
(1.8)
+ zzzzzzzzzzzzzzzzzz,
bbb = tttttttttttttttttt,
hh = vvvvvvvvvvv.
\begin{equation}\label{E:a4}
\begin{split}
xxxxx &= yyyyyyyyyyyyyy\\
&\quad + zzzzzzzzzzzzzzzzzz,\\
bbb &= tttttttttttttttttt,\\
hh &= vvvvvvvvvvv.
\end{split}
\end{equation}
Wzory – katalog
3
aaaaaaaaaaaaaaaaaaaaaaa = bbbbbbbbbbbbb,
(1.9)
bbbb = xxxxxx,
ccccc = yyyyyyy,
(1.10)
dddddddd = zzzzz.
\begin{align}
\begin{split}
aaaaaaaaaaaaaaaaaaaaaaa &= bbbbbbbbbbbbb,\\
bbbb &= xxxxxx,
\end{split}\label{Ea5}\\
\begin{split}
ccccc &= yyyyyyy,\\
dddddddd &= zzzzz.\label{E:a6}
\end{split}
\end{align}
aaaaaaaaaaaaaaaaaaaaaaa = bbbbbbbbbbbbb,
(1.11)
bbbbbbbbbbbbbbbbbbbbbbbb = xxxxxx,
ccccc = yyyyyyy,
(1.12)
dddddddd = zzzzz.
\begin{gather}
\begin{split}
& aaaaaaaaaaaaaaaaaaaaaaa = bbbbbbbbbbbbb,\\
& bbbbbbbbbbbbbbbbbbbbbbbb = xxxxxx,
\end{split} \label{E:a7}\\
\begin{split}
& ccccc = yyyyyyy,\\
& dddddddd = zzzzz.
\end{split}\label{E:a8}
\end{gather}
aa = bbb,
dd = ee
hh = ii,
\begin{align*}
aa &= bbb, & dd &= ee
hh &= ii,
& ll &= kkk
\end{align*}
ll = kkk
&
&
&\text{(by Lemma 2),}\\
&\text{(by \eqref{E:a8}).}
(by Lemma 2),
(by (1.12)).
4
Wzory – katalog
(1.13)
aa = bbb,
(1.14)
hh = ii,
\begin{alignat}{3}
aa &= bbb,\quad\ &
hh &= ii,
&
\end{alignat}
(1.15)
dd = ee
dd &= ee
ll &= kkk\quad\
aa = bbb,
hh = ii,
aa = bbb,
(1.16b)
hh = ii,
(1.17b)
&
&
(by Lemma 2),
(by (1.12)).
&\text{(by Lemma 2),}\\
&\text{(by \eqref{E:a8}).}
dd = ee
(by Lemma 2),
ll = kkk
&
&
(by (1.12)).
&\text{(by Lemma 2),} \label{E:a9}\\
&\text{(by \eqref{E:a8}).\label{E:a10}}
ll = kkk
\begin{subequations}\label{E:suba}
\begin{alignat}{3}
aa &= bbb,\quad\ & dd &= ee
hh &= ii,
& ll &= kkk\quad\
\end{alignat}
\end{subequations}
(1.17a)
&
&
dd = ee
\begin{equation}\label{E:a11}
\begin{alignedat}{3}
aa &= bbb,\quad\ & dd &= ee
hh &= ii,
& ll &= kkk\quad\
\end{alignedat}
\end{equation}
(1.16a)
(by Lemma 2),
ll = kkk
(by (1.12)).
&\text{(by Lemma 2),} \label{E:suba1}\\
&\text{(by \eqref{E:a8}).} \label{E:suba2}
aaaaaaaaaaaaaaaaaaaaaaa = bbbbbbbbbbbbb,
bbbbbbbbbbbbbbbbbbbbbbbb = xxxxxx,
ccccc = yyyyyyy,
dddddddd = zzzzz.
\begin{subequations}\label{E:subg}
\begin{gather}
\begin{split}
& aaaaaaaaaaaaaaaaaaaaaaa = bbbbbbbbbbbbb, \\
& bbbbbbbbbbbbbbbbbbbbbbbb = xxxxxx,
\end{split}\label{E:subg1}\\
\begin{split}
& ccccc = yyyyyyy,\\
& dddddddd = zzzzz.
\end{split}\label{E:subg2}
\end{gather}
\end{subequations}
Wzory – katalog
2
5
Jeden wzór
(2.1) aaaaaaaaaaaaaaaaaaaaaaaaaaaa
+ bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
≤ dddddddddddddddddddddddddd.
\begin{multline}\label{E:m1}
aaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
+ bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb\\
\le dddddddddddddddddddddddddd.
\end{multline}
(2.2) aaaaaaaaaaaaaaaaaaaaaaaaaaaa
+ bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb + dddddddddd − ggggggggg
× eeeeeeeeeeeeeeeeeeeeeeee
≤ dddddddddddddddddddddddddd.
\begin{multline}\label{E:m2}
aaaaaaaaaaaaaaaaaaaaaaaaaaaa\\
\shoveleft{+ bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb + dddddddddd - ggggggggg}\\
\shoveright{\times eeeeeeeeeeeeeeeeeeeeeeee}\\
\le dddddddddddddddddddddddddd.
\end{multline}
(2.3) aaaaaaaaaaaaaaaaa + xxxxxxxxxxxx
< bbbbbbbbbbbbbbbbbbbbbbb
+ [dddddd + eeeeeeee][hhhhhhhh − ggggggggggg]
< ccccccccccccccccccc.
\begin{multline}\label{E:m3}
aaaaaaaaaaaaaaaaa + xxxxxxxxxxxx\\
\begin{aligned}
&< bbbbbbbbbbbbbbbbbbbbbbb\\
&\quad + [dddddd + eeeeeeee][hhhhhhhh - ggggggggggg]\\
&< ccccccccccccccccccc.
\end{aligned}
\end{multline}
6
Wzory – katalog
1
(2.4)
Pascal3 = 1 2 1
1331
\begin{equation}\label{E:pasc}
\mathrm{Pascal}_{3} = \begin{gathered}
1\\
1\ 2\ 1\\
1\ 3\ 3\ 1
\end{gathered}
\end{equation}
1
1
121
121
1331
1
1
121
121
\begin{equation*}
\begin{gathered}[b]
1\\
1\ 2\ 1
\end{gathered}
\qquad
\begin{gathered}[b]
1\\
1\ 2\ 1\\
1\ 3\ 3\ 1
\end{gathered}
\end{equation*}
1331
\begin{equation*}
\begin{gathered}[t]
1\\
1\ 2\ 1
\end{gathered}
\qquad
\begin{gathered}[t]
1\\
1\ 2\ 1\\
1\ 3\ 3\ 1
\end{gathered}
\end{equation*}
Wzory – katalog
(2.5)
7
A = zt = ztuv + [f1 (a, b, c, d, e, f, g, h),
f2 (a, b, c, d, e, f, g, h),
f3 (a, b, c, d, e, f, g, h)]
= cccccccccccccc
\begin{align}\label{E:top}
A &= \begin{aligned}[t]
zt = ztuv + [&f_1(a,b,c,d,e,f,g,h),\\
&f_2(a,b,c,d,e,f,g,h),\\
&f_3(a,b,c,d,e,f,g,h)]
\end{aligned}\\
&= cccccccccccccc\notag
\end{align}
xxxxxxx + [f1 (a, b, c, d, e, f, g, h),
f2 (a, b, c, d, e, f, g, h),
f3 (a, b, c, d, e, f, g, h)] = tttttttttttttttttttttttttt
= bbbbbbbbbbbbbb.
\begin{align*}
\begin{aligned}[b]
xxxxxxx + [&f_1(a,b,c,d,e,f,g,h),\\
&f_2(a,b,c,d,e,f,g,h),\\
&f_3(a,b,c,d,e,f,g,h)]
\end{aligned}
&= tttttttttttttttttttttttttt\\
&= bbbbbbbbbbbbbb.
\end{align*}
(2.6)
xxxxx = yyyyyyyyyyyyyy + [eeee
× zzzzzzzzzzzzzzzzzz]
= tttttttttttttttttt
= vvvvvvvvvvv.
\begin{align}\label{E:popch}
xxxxx &= \begin{aligned}[t]
yyyyyyyyyyyyyy + [&eeee\\
&\times zzzzzzzzzzzzzzzzzz]
\end{aligned}\\
&= tttttttttttttttttt \notag\\
&= vvvvvvvvvvv.\notag
\end{align}
8
(2.7)
Wzory – katalog
xxxxx = yyyyyyyyyyyyyy + [eeee
× zzzzzzzzzzzzzzzzzz]
(note that we have not used the full strength of (H) here, but only the concavity of f )
= tttttttttttttttttt
= vvvvvvvvvvv.
\begin{align}
xxxxx &= \begin{aligned}[t]
yyyyyyyyyyyyyy + [&eeee
\\
&\times zzzzzzzzzzzzzzzzzz]
\end{aligned}\label{E:inter} \\
\intertext{(\emph{note that we have not used the full strength
of $(H)$ here, but only the concavity of $f$})}
&= tttttttttttttttttt \notag\\
&= vvvvvvvvvvv.\notag
\end{align}
3
Użycie makr
1
2 (u
+ v)
2
v+z
n
Y
2 u n
g + rh
+
Ai +
.
n+1
v
i=1
u+
=
\[
\bigl(\tfrac{1}{2}(u+v)\bigr)^2
= \frac{u + \dfrac {v+z}{g+rh}}{n+1}
+ \Bigl(\prod_{i=1}^n A_i\Bigr)^2
+ \biggl(\frac{u}{v}\biggr)^n.
\]
(C)
α
f (x) =
(p
2/sin x if x ∈ (0, π),
0
otherwise.
3
\[
f(x)\overset{\alpha} =
\begin{cases}
\sqrt[3]{2/\!\sin x} &\text{if $x \in (0, \pi)$,}\\
0
&\text{otherwise.}
\end{cases}
\tag{C}
\]
Wzory – katalog
9
d2
a+b+c+d
A −−−−−−→ B −−→ C D −→ gdeg E.
abc
Makra:
\usepackage(amssymb}
\newcommand{\arr}{\xrightarrow}
\newcommand{\ssquare}{\mathbin{\square}}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\BB}{\mathbf{B}}
\newcommand{\frC}{\mathfrak{C}}
\newcommand{\biD}{\boldsymbol{D}}
\DeclareMathOperator{\gdeg}{\mathsf{gdeg}}
Kod:
\[
\bA \arr{a+b+c+d} \BB \arr[abc]{} \frC \ssquare \biD \arr{d^2} \gdeg E.
\]
X0 m + n YY
=
akl
k+l
k<m, l<n
k+l+m=3
2k−l+n≤7
Makra:
\newcommand{\prsum}{\sideset{}{’}\sum}
\newcommand{\dprod}{\operatorname*{\prod\prod}}
Kod:
\prsum_{k<m,\,l<n} \binom{m+n}{k+l}
= \dprod_{\substack{k+l+m=3\\2k-l+n\le 7}} a_{kl}
w∗ -lim an
n→∞
=
ha, bi
hc, ai
ha, ci
hb, ci
Makra:
\newcommand{\wstlim}{\mathop{w^*\textup{-lim}}}
\def\<#1>{\langle#1\rangle}
Kod:
\[
\wstlim_{n\to\infty} a_{n} =
\begin{pmatrix}
\<a,b> & \<a,c>\\
\<c,a> & \<b,c>
\end{pmatrix}
\]