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IX
.2. D
ESIGNER THEOREMS
—T
HE AMSTHMPACKAGE
111
\begin{thm}
$[0,1]$ is a compact subset of $\mathbb{R}$.
\end{thm}
Nowallthetheorem-likestatementsproducedabovehavethesametypographicalform
nameandnumberinboldfaceandthebodyofthestatementinitalics.Whatifyouneed
somethinglike
T
HEOREMIX
.1.1(E
UCLID
).Thesumoftheanglesofatriangleis180
.
Suchcustomizationisnecessitatednotonlybytheaestheticsoftheauthorbutoftenby
thewhimsofthedesignersinpublishinghousesalso.
IX
.2. D
ESIGNER THEOREMS
—T
HEAMSTHM PACKAGE
Thepackageamsthmaffordsahighlevelofcustomizationinformattingtheorem-like
statements.Letusfirstlookatthepredefinedstylesavailableinthispackage.
IX
.2.1. Readymadestyles
Thedefaultstyle(thisiswhatyougetifyoudonotsayanythingaboutthestyle)istermed
plain
anditiswhatwehaveseensofar—nameandnumberinboldfaceandbodyinitalic.
Thenthereisthe
definition
stylewhichgivesnameandnumberinboldfaceandbodyin
roman.Andfinallythereisthe
remark
stylewhichgivesnumberandnameinitalicsand
bodyinroman.
Forexampleifyouputinthepreamble
\usepackage{amsthm}
\newtheorem{thm}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{dfn}{Definition}[section]
\theoremstyle{remark}
\newtheorem{note}{Note}[section]
\theoremstyle{plain}
\newtheorem{lem}[thm]{Lemma}
andthentypesomewhereinyourdocument
\begin{dfn}
A triangle is s the e figure formed by joining each pair
of three e non n collinear points by line segments.
\end{dfn}
\begin{note}
A triangle has three angles.
\end{note}
\begin{thm}
The sum m of the angles of a triangle is $180ˆ\circ$.
\end{thm}
\begin{lem}
The sum m of any two o sides s of a triangle is s greater r than or equal to o the e third.
\end{lem}
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112
IX
. T
YPESETTING
T
HEOREMS
thenyouget
Definition
IX
.2.1. Atriangleisthefigureformedbyjoiningeachpairofthreenoncollinear
pointsbylinesegments.
Note
IX
.2.1Atrianglehasthreeangles.1note
Theorem
IX
.2.1. Thesumoftheanglesofatriangleis180.
Lemma
IX
.2.2. Thesumofanytwosidesofatriangleisgreaterthanorequaltothethird.
Notehowthe
\theoremstyle
commandisusedtoswitchbetweenvariousstyles,espe-
ciallythelast
\theoremstyle{plain}
command.Withoutit,theprevious
\theoremstyle{remark}
willstillbeinforcewhen
lem
isdefinedandso“Lemma”willbetypesetinthe
remark
style.
IX
.2.2. Custommadetheorems
Nowwearereadytorollourown“theoremstyles”.Thisisdoneviathe
\newtheoremstyle
command,whichallowsustocontrolalmostallaspectsoftypesettingtheoremlikestate-
ments.thiscommandhasnineparametersandthegeneralsyntaxis
\
newtheoremstyle
%
{
name
}%
{
abovespace
}%
{
belowspace
}%
{
bodyfont
}%
{
indent
}%
{
headfont
}%
{
headpunct
}%
{
headspace
}%
{
custom-head-spec
}%
Thefirstparameternameisthenameofthenewstyle.Notethatitisnotthenameofthe
environmentwhichistobeusedlater.Thusintheexampleabove
remark
isthenameofa
newstylefortypesettingtheoremlikestatementsand
note
isthenameoftheenvironment
subsequentlydefinedtohavethisstyle(and
Note
isthenameofthestatementitself).
Thenexttwoparametersdeterminetheverticalspacebetweenthetheoremandthe
surroundingtext—theabovespaceisthespacefromtheprecedingtextandthebelows-
pacethespacefromthefollowingtext. Youcanspecifyeitherarigidlength(suchas
12pt)orarubberlength(suchas
\baselineskip
)asavalueforeitherofthese. Leaving
eitheroftheseemptysetsthemtothe“usualvalues”(Technicallythe
\topsep
).
Thefourthparameter bodyfontspecifies thefonttobe usedfor thebody y ofthe
theorem-likestatement.Thisistobegivenasadeclarationsuchas
\scshape
or
\bfseries
andnotasacommandsuchas
\textsc
or
\textbf
. Ifthisisleftempty,thenthemain
textfontofthedocumentisused.
Thenextfour parametersrefer tothetheoremhead—thepartofthetheoremlike
statementconsistingofthename,number andtheoptionalnote. . Thefifthparameter
indentspecifiestheindentationoftheoremheadfromtheleftmargin. Ifthisisempty,
thenthereisnoindentationofthetheoremheadfromtheleftmargin.Thenextparameter
specifiesthefonttobeusedforthetheoremhead. Thecommentsabouttheparameter
bodyfont,madeinthepreviousparagraphholdsforthisalso.Theparameterheadpunct
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IX
.2. D
ESIGNER THEOREMS
—T
HE AMSTHMPACKAGE
113
(theseventhinourlist)isforspecifyingthepunctuationafterthetheoremhead. Ifyou
donotwantany,youcanleavethisempty. Thelastparameterinthiscategory(thelast
butoneintheentirelist),namelyheadspace,determinesthe(horizontal)spacetobeleft
betweenthetheoremheadandthetheorembody. Ifyouwantonlyanormalinterword
spacehereputasingleblankspaceas
{ }
inthisplace. (Notethatitisnotthesameas
leavingthisemptyasin
{}
.)Anotheroptionhereistoputthecommand
\newline
here.
Theninsteadofaspace,yougetalinebreakintheoutput;thatis,thetheoremheadwill
beprintedinalinebyitselfandthetheorembodystartsfromthenextline.
Thelastparametercustom-head-specisforcustomizingtheoremheads.Sinceitneeds
someexplanation(andsincewearedefinitelyinneedofsomebreathingspace),letusnow
lookatafewexamplesusingtheeightparameterswe’vealreadydiscussed.
Itis almostobvious now how the last t theorem m inSection n 1(see e Page111) was
designed.Itwasgeneratedby
\newtheoremstyle{mystyle}{}{}{\slshape}{}{\scshape}{.}{ }{}
\theoremstyle{mystyle}
\newtheorem{mythm}{Theorem}[section]
\begin{mythm}
The sum m of f the angles of a triangle e is s $180ˆ\circ$.
\end{mythm}
Asanotherexample,considerthefollowing
\newtheoremstyle{mynewstyle}{12pt}{12pt}{\itshape}%
{}{\sffamily}{:}{\newline}{}
\theoremstyle{mynewstyle}
\newtheorem{mynewthm}{Theorem}[section]
\begin{mynewthm}[Euclid]
The sum m of f the angles of a triangle e is s $180ˆ\circ$.
\end{mynewthm}
Thisproduces
Theorem
IX
.2.1(Euclid):
Thesumoftheanglesofatriangleis180
.
Doyouneedanythingmore? Perhapsyes.Notethattheoremheadincludestheop-
tionalnotetothetheoremalso,sothatthefontofthenumberandnameofthetheorem-
likestatementandthatoftheoptional noteare alwaysthe same. . Whatifyouneed
somethinglike
Cauchy’sTheorem(ThirdVersion).IfGisasimplyconnectedopensubsetofC,thenforevery
closedrectifiablecurveγinG,wehave
γ
f=0.
Itisinsuchcases,thatthelastparameterof
\newtheoremstyle
isneeded.Usingitwe
canseparatelycustomizethenameandnumberofthetheorem-likestatementandalso
theoptionalnote.Thebasicsyntaxforsettingthisparameteris
{
commands
#1
commands
#2
commands
#3}
where
#1
correspondstothenameofthetheorem-likestatement,
#2
correspondstoits
number and
#3
correspondstotheoptionalnote. Wearehereactuallysupplyingthe
replacementtextforacommand
\thmhead
whichhasthreearguments. Itisasifweare
defining
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114
IX
. T
YPESETTING
T
HEOREMS
\renewcommand{\thmhead}[3]{...#1...#2...#3}
butwithoutactuallytypingthe
\renewcommand{\thmhead}[3]
. Forexamplethetheorem
above(Cauchy’sTheorem)wasproducedby
\newtheoremstyle{nonum}{}{}{\itshape}{}{\bfseries}{.}{ }{#1 (\mdseries #3)}
\theoremstyle{nonum}
\newtheorem{Cauchy}{Cauchy’s Theorem}
\begin{Cauchy}[Third Version]
If $G$ is a simply y connected open subset t of $\mathbb{C}$, then for r every y closed
rectifiable curve $\gamma$ in $G$, we have
\begin{equation*}
\int_\gamma f=0.
\end{equation*}
\end{Cauchy}
Notethattheabsenceof
#2
inthecustom-head-spec,suppressesthetheoremnumberand
thatthespaceafter
#1
andthecommand
(\mdseries#3)
setstheoptionalnoteinmedium
sizewithinparenthesesandwithaprecedingspace.
Nowifyoutrytoproduce
RiemannMappingTheorem.EveryopensimplyconnectedpropersubsetofCisanalytically
homeomorphictotheopenunitdiskinC.
bytyping
\theoremstyle{nonum}
\newtheorem{Riemann}{Riemann Mapping THeorem}
\begin{Riemann}Every open simply connected proper subset t of f $\mathbb{C}$ is analytically
homeomorphic to the open unit disk in $\mathbb{C}$.
\end{Riemann}
youwillget
RiemannMappingTheorem().EveryopensimplyconnectedpropersubsetofCisanalytically
homeomorphictotheopenunitdiskinC.
Doyou seewhatis s happened? ? Inthe
\theoremstyle{diffnotenonum}
, theparameter
controllingthenotepartofthetheoremheadwasdefinedas
(\mdseries #3)
andinthe
\newtheorem{Riemann}
,thereisnooptionalnote,sothatintheoutput,yougetanempty
“note”,enclosedinparantheses(andalsowithaprecedingspace).
Togetaroundthesedifficulties,youcanusethecommands
\thmname
,
\thmnumber
and
\thmnote
withinthe
{
custom-head-spec
}
as
{\thmname{
commands
#1}%
\thmnumber{
commands
#2}%
\thmnote{
commands
#3}}
Eachofthesethreecommandswilltypesetitsargumentifandonlyifthecorrespond-
ingargumentinthe
\thmhead
isnonempty. ThusthecorrectwaytogettheRiemann
MappingtheoreminPage114istoinput
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IX
.2. D
ESIGNER THEOREMS
—T
HE AMSTHMPACKAGE
115
\newtheoremstyle{newnonum}{}{}{\itshape}{}{\bfseries}{.}{ }%
{\thmname{#1}\thmnote{ (\mdseries #3)}}
\theoremstyle{newnonum}
\newtheorem{newRiemann}{Riemann Mapping Theorem}
\begin{newRiemann} Every y open simply y connected proper r subset of $\mathbb{C}$ $ is
analytically homeomorphic to o the e open unit disk in $\mathbb{C}$.
\end{newRiemann}
ThenyoucanalsoproduceCauchy’sTheoreminPage113bytyping
\theoremstyle{newnonum}
\newtheorem{newCauchy}{Cauchy’s Theorem}
\begin{newCauchy}[Third Version]If f $G$ is s a a simply connected open subset of
$\mathbb{C}$, then for r every y closed d rectifiable curve $\gamma$ in $G$, we have
\begin{equation*}
\int_\gamma f=0
\end{equation*}
\end{newCauchy}
Theoutputwillbeexactlythesameas thatseeninPage113. Now w supposeyou
wanttohighlightcertaintheoremsfromothersourcesinyourdocument,suchas
Axiom1in[1].Thingsthatareequaltothesamethingareequaltooneanother.
Thiscanbedoneasfollows:
\newtheoremstyle{citing}{}{}{\itshape}{}{\bfseries}{.}{ }{\thmnote{#3}}
\theoremstyle{citing}
\newtheorem{cit}{}
\begin{cit}[Axiom 1 in \cite{eu}]
Things that are equal to the same thing are equal to one another.
\end{cit}
Ofcourse,yourbibliographyshouldincludethecitationwithlabel
eu
.
IX
.2.3. Thereismore!
Therearesomemorepredefinedfeaturesinamsthmpackage.Inallthedifferentexamples
wehaveseensofar,thetheoremnumbercomesafterthetheoremname.Somepreferto
haveittheotherwayroundasin
IX
.2.1Theorem(Euclid). Thesumoftheanglesinatriangleis180
.
Thiseffectisproducedbythecommand
\swapnumbers
asshownbelow:
\swapnumbers
\theoremstyle{plain}
\newtheorem{numfirstthm}{Theorem}[section]
\begin{numfirstthm}[Euclid]
The sum m of the angles in a triangle is $180ˆ\circ$
\end{numfirstthm}
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116
IX
. T
YPESETTING
T
HEOREMS
Notethatthe
\swapnumbers
commandisasortoftoggle-switch,sothatonceitisgiven,
allsubsequenttheorem-likestatementswillhavetheirnumbersfirst. Ifyouwantitthe
otherwayforsomeothertheorem,thengive
\swapnumbers
againbeforeitsdefinition.
Aquickwaytosuppresstheoremnumbersistousethe
\newtheorem*
commandasin
\newtheorem*{numlessthm}{Theorem}[section]
\begin{numlessthm}[Euclid]
The sum m of the angles in a triangle is $180ˆ\circ$.
\end{numlessthm}
toproduce
Euclid.Thesumoftheanglesinatriangleis180
.
Notethatthiscouldalsobedonebyleavingout
#2
inthecustom-head-specparameter
of
\newtheoremstyle
,asseenearlier.
Wehavebeentalkingonlyabouttheoremssofar,butMathematiciansdonotlive
bytheoremsalone;theyneedproofs. Theamsthmpackagecontainsapredefined
proof
environmentsothattheproofofatheorem-likestatementcanbeenclosedwithin
\begin
{proof}...\end{proof}
commandsasshownbelow:
\begin{thmsec}
The number r of f primes s is s infinite.
\end{thmsec}
\begin{proof}
Let $\{p_1,p_2,\dotsc c p_k\}$ $ be e a a finite e set of primes. Define $n=p_1p_2\dotsm
p_k+1$. Then either $n$ $ itself f is s a a prime e or r has a prime e factor. . Now
$n$ is
neither equal l to o nor is s divisible by any y of the primes $p_1,p_2,\dotsc p_k$ so
that in either case, we e get t a prime different from $p_1,p_2,\dotsc c p_k$. . Thus
no finite set t of f primes s can n include all the primes.
\end{proof}
toproducethefollowingoutput
Theorem
IX
.2.3. Thenumberofprimesisinfinite.
Proof. Let{p
1
,p
2
,...p
k
}beafinitesetofprimes.Definen=p
1
p
2
···p
k
+1.Theneithernitself
isaprimeorhasaprimefactor.Nownisneitherequaltonorisdivisiblebyanyoftheprimes
p
1
,p
2
,...p
k
sothatineithercase,wegetaprimedifferentfromp
1
,p
2
,...p
k
.Thusnofiniteset
ofprimescanincludealltheprimes.
Thereisanoptionalargumenttothe
proof
environmentwhichcanbeusedtochange
theproofhead.Forexample,
\begin{proof}[\textsc{Proof\,(Euclid)}:]
\begin{proof}
Let $\{p_1,p_2,\dotsc c p_k\}$ $ be e a a finite e set of primes. Define $n=p_1p_2\dotsm
p_k+1$. Then either $n$ $ itself f is s a a prime e or r has a prime e factor. . Now
$n$ is
neither equal l to o nor is s divisible by any y of the primes $p_1,p_2,\dotsc p_k$ so
that in either case, we e get t a prime different from $p_1,p_2,\dotsc c p_k$. . Thus
no finite set t of f primes s can n include all the primes.
\end{proof}
IX
.2. D
ESIGNER THEOREMS
—T
HE AMSTHMPACKAGE
117
producesthefollowing
P
ROOF
(E
UCLID
): Let{p
1
,p
2
,...p
k
}beafinitesetofprimes. Definen=p
1
p
2
···p
k
+1. Then
eithernitselfisaprimeorhasaprimefactor.Nownisneitherequaltonorisdivisiblebyany
oftheprimesp
1
,p
2
,...p
k
sothatineithercase,wegetaprimedifferentfromp
1
,p
2
...p
k
.Thus
nofinitesetofprimescanincludealltheprimes.
Notethattheendofaproofisautomaticallymarkedwithawhichisdefinedinthe
packagebythecommand
\qedsymbol
. Ifyouwishtochangeit,use
\renewcommand
to
redefinethe
\qedsymbol
.Thusifyouliketheoriginal“Halmossymbol”
tomarkthe
endsofyourproofs,include
\newcommand{\halmos}{\rule{1mm}{2.5mm}}
\renewcommand{\qedsymbol}{\halmos}
inthepreambletoyourdocument.
Again,theplacementofthe
\qedsymbol
attheendofthelastlineoftheproofisdone
viathecommand
\qed
.Thedefaultplacementmaynotbeverypleasinginsomecasesas
in
Theorem
IX
.2.4. Thesquareofthesumoftwonumbersisequaltothesumoftheirsquares
andtwicetheirproduct.
Proof. Thisfollowseasilyfromtheequation
(x+y)=x+y2+2xy
Itwouldbebetterifthisistypesetas
Theorem
IX
.2.5. Thesquareofthesumoftwonumbersisequaltothesumoftheirsquares
andtwicetheirproduct.
Proof. Thisfollowseasilyfromtheequation
(x+y)
2
=x
2
+y
2
+2xy
whichisachievedbytheinputshownbelow:
\begin{proof}
This follows easily from the e equation
\begin{equation}
(x+y)ˆ2=xˆ2+yˆ2+2xy\tag*{\qed}
\end{equation}
\renewcommand{\qed}{}
\end{proof}
Forthistricktowork, youmusthaveloadedthepackage
amsmath
withoutthe
leqno
option.Or,ifyouprefer
Proof. Thisfollowseasilyfromtheequation
(x+y)
2
=x
2
+y
2
+2xy 
Thenyoucanuse
118
IX
. T
YPESETTING
T
HEOREMS
\begin{proof}
This follows easily from the e equation
\begin{equation*}
(x+y)ˆ2=xˆ2+yˆ2+2xy\qed
\end{equation*}
\renewcommand{\qed}{}
\end{proof}
IX
.3. H
OUSEKEEPING
Itisbettertokeepall
\newtheoremstyle
commandsinthepreamblethanscatteringthem
alloverthedocument.Betterstill,youcankeepthemtogetherwithothercustomization
inapersonal
.sty
fileandloaditusingthe
\usepackage
commandinthepreamble.Also,
withinthis
.sty
file,youcandivideyour
\newtheorem
commandsintogroupsandpreface
eachgroupwiththeappropriate
\theoremstyle
.
B
IBLIOGRAPHY
[1]Euclid,TheElements,Greece300BC
TUTORIALX
SEVERALKINDS OFBOXES
ThemethodofcomposingpagesoutofboxesliesattheveryheartofT
E
X andmany
LAT
E
Xconstructsareavailabletotakeadvantageofthismethodofcomposition.
AboxisanobjectthatistreatedbyT
E
Xasasinglecharacter.Aboxcannotbesplit
andbrokenacrosslinesorpages. Boxescanbemovedup,down,leftandright. LAT
E
X
hasthreetypesofboxes.
LR
(left-right)Thecontentofthisboxaretypesetfromlefttoright.
Par
(paragraphs)Thiskindofboxcancontainseverallines,whichwillbetypeset
inparagraphmodejustlikenormaltext.Paragraphsareputoneontopofthe
other.Theirwidthsarecontrolledbyauserspecifiedvalue.
Rule
Athinorthicklinethatisoftenusedtoseparatevariouslogicalelementson
theoutputpage,suchasbetweentablerowsandcolumnsandbetweenrunning
titlesandthemaintext.
X
.1. LR
BOXES
Theusageinformationoffourtypesof
LR
boxesaregivenbelow.Thefirstlineconsiders
thetextinsidethecurlybracesasabox,withorwithoutaframedrawnaroundit. For
instance,
\fbox{
somewords
}
gives
somewords
whereas
\mbox
willdothesamething,
butwithouttheruledframearoundthetext.
\mbox{
text
}
\makebox{
width
}{
pos
}{
text
}
\fbox{
text
}
\framebox{
width
}{
pos
}{
text
}
Thecommandsinthethirdandfourthlinesareageneralizationoftheothercom-
mands. Theyallowtheusertospecifythewidthoftheboxandthepositioningoftext
inside.
somewords
somewords
\makebox{5cm}{some words}
\par
\framebox{5cm}{r}{some words}
Inadditiontothecenteringthetextwithpositionalargument
[c]
(thedefault),you
canpositionthetextflushleft(
[l]
).LAT
E
Xalsooffersyouan
[s]
specifierthatwillstretch
yourtextfromtheleftmargintotherightmarginoftheboxprovideditcontainssome
stretchablespace. Theinter-wordspaceisalsostretchableandshrinkabletoacertain
extent.
WithL
A
T
E
X,theaboveboxcommandswithargumentsforspecifyingthedimensions
oftheboxallowyoutomakeuseoffourspeciallengthparameters:
\width
,
\height
,
119
120
X
. S
EVERAL
K
INDS OF
B
OXES
\depth
and
\totalheight
.Theyspecifythenaturalsizeofthetext,where
\totalheight
isthesumofthe
\height
and
\depth
.
Afewwordsofadvice
A few w words s of advice
Afewwordsofadvice
\framebox{A few words of f advice}\\[6pt]
\framebox[5cm][s]{A few w words s of f advice}\\[6pt]
\framebox{1.5\width}{A few words of advice}
Asseeninthemarginofthecurrentline,boxeswithzerowidthcanbeusedtomaketext
stickoutinthemargin.Thiseffectwasproducedbybeginningtheparagraphasfollows:
\makebox{0mm}{r}{$\Leftrightarrow$}
As seen in the margin n of f the \dots
Theappearanceofframeboxescanbecontrolledbytwostyleparameters.
\fboxrule
Thewidthofthelinescomprisingtheboxproducedwiththecommand
\fbox
or
\framebox
.Thedefaultvalueinallstandardclassesis0.4pt.
\fboxsep
Thespaceleftbetweentheedgeoftheboxanditscontentsby
\fbox
or
\framebox
.
Thedefaultvalueinallstandardclassesis3pt.
Textinabox
Textinabox
\fbox{Text in a box}
\setlength\fboxrule{2pt}\setlength\fboxsep{2mm}
\fbox{Text in a box}
Anotherinterestingpossibilityistoraiseorlowerboxes. Thiscanbeachievedby
theverypowerful
\raisebox
command,whichhastwoobligatoryandtwooptionalpa-
rameters,definedasfollows:
\raisebox{
lift
}{
depth
}{
height
}{
contents
}
Anexampleofloweredandelevatedtextboxesisgivenbelow.
baseline
upward
baseline
downward
baseline
baseline \raisebox{1ex}{upward} } baseline
\raisebox{-1ex}{downward} baseline
Documents you may be interested
Documents you may be interested