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MISCELLANEOUSXMAJTHEMATICSFEATURES&[98G#\setcounter.`T*o makethisalittlemoreconvenient,*theamsmathpackqagepro- #videsL1acommand\numberwithin.nT*ohaveL1equationnumbGeringL1tiedtosection#numbGering,WwithVautomaticresetoftheequationcounter,thecommandwould#bGe/#\numberwithin{equation}{section}d#2.11.2KCrossTreferencestoequationn9umbQers#T*opbmakecross-referencestoequationseasier,w&an\eqrefcommandisprovided.#ThisautomaticallysuppliestheparenthesesaroundtheequationnumbGer,"and#adds anitaliccorrectionifnecessary*. To refertoanequationthatwaslabGeled#withUUthelabGele:baset,theusagewouldbe\eqref{e:baset}.?#2.11.3KSubQordinateTn9umberingTsequences#The^amsmathpackqageprovidesalsoasubequationsenvironmenttomakeit#easytonumbGerequationsinaparticulargroupwithasubordinatenumbering#scheme.qF*orUUexample/#\begin{subequations}#...#\end{subequations}d#causesHallnumbGeredHequationswithinthatpartofthedocumenttobenumbered#(4.9a)](4.9b)(4.9c):::]~,iftheprecedingnumbGered]equationwas(4.8).-A]H\label#command؃immediatelyafter\begin{subequations}willproGducea\refofthe#parentnumbGer4.9,¬4.9a.ThecountersusedbythesubGequationsenviron-#mentOareparentequationandequationand\addtocounter,\setcounter,#\value,{etc.,cantbGeappliedasusualtothosecounternames.T*ogetanything#otherthanlowercaselettersforthesubGordinatenumbers,usestandardL5ffA͉TU>'ExX#methoGdsqforchangingnumbGeringstyle[5,8x6.3,xC.8.4].UF*orqexample,rede ning#\theequationUUasfollowswillproGduceromannumerals.#\begin{subequations}#\renewcommand{\theequation}{\theparentequation?\roman{equation}}#...(d#|3|!#Miscellaneousmathematicsfeatures(amsmathpack@age)!#3.1=1MatricesT#The=amsmathpackqageprovidessomeenvironmentsformatricesbGeyondthebasic#array DzenvironmentofL5ffA͉TU>'ExX.Thepmatrix,6bmatrix,Bmatrix,vmatrixand#Vmatrixhave(respGectively)(),g[],fg,jj,andkkdelimitersbuiltin. jF*or#namingo consistencythereisamatrixenvironmento sansdelimiters.Thisisnot#entirelyJredundantwiththearrayenvironment;řthematrixenvironmentsall 5s\ȍ8Y10&Z3. MISCELLANEOUSXMAJTHEMATICSFEATURES8GYuse-moreeconomicalhorizontalspacingthantheratherproGdigalspacingofthe Yarrayenvironment.Also,ٙunlikethearrayenvironment,ٙyoudon'thavetoYgivecolumnspGeci cationsforanyofthematrixenvironments;bydefaultyouYcanhaveupto10centeredcolumns.^1 .(IfyouneedleftorrightalignmentinaYcolumnUUorotherspGecialformatsyoumustresorttoarray.)hT*o3proGduceasmallmatrixsuitableforuseintext,*thereisasmallmatrixYenvironmentS(e.g.,b @ab\q Zc]d2BbǙ)thatcomescloserto ttingwithinasingletextlineYthanUXanormalmatrix.sDelimitersmustbGeprovided;therearenop,b,v,VUversionsYofUUsmallmatrix.qTheabGoveUUexamplewasproducedbyY\bigl(?\begin{smallmatrix}ca&b\\?c&dY\end{smallmatrix}?\bigr)h\hdotsfor{hnumb}'eri}xproGducesarowofdotsinamatrixspanningthegivenYnumbGerUUofcolumns.qF*orexample,)gdzai/bcdzpe T:T:T:T:T:T:T::F\begin{matrix}?a&b&c&d\\Fe&\hdotsfor{3}?\end{matrix}hThespacingofthedotscanbGevqariedthroughuseofasquare-bracketoption,YforSexample,Sy\hdotsfor[1.5]{3}.qThenumbGerSinsquarebracketsSwillbeusedYasUUamultiplier(i.e.,thenormalvqalueis1.0).&eV0BBfi@dŅKD1|st{a12xt2/:::&~a1n mtnma21xt1oD2|st/:::&~a2n mtnZ:S:::::::::::::::::::::ian1 mt1 an2 mt2/:::.Dnq~tVFy1FyCFyCfiFyAPV;Y(3.1)%_Y\begin{pmatrix}?D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\Y-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\Y\hdotsfor[2]{4}\\Y-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}Y3.2s1MathTspacingcommandsȍYTheamsmathpackqageslightlyextendsthesetofmathspacingcommands,FPasYshownbGelow.BoththespGelled-outandabbreviatedformsofthesecommandsYareUUrobust,andtheycanalsobGeusedoutsideofmath '31_Abbrev.$SpGelledUUoutExampleAbbrev.4SpGelledUUoutExampleYff]fd\,5$\thinspacez F\!\negthinspace1̈́6̈́5n6Y\:5$\medspacez~3\negmedspace1̈́6̈́45z!\;5$\thickspacez'P\negthickspace1̈́6̈́4RO45$\quadz`95$\qquadz`:Y'uiffv J= "5-:1LMoreprecisely:KThemaximÎumnumb'ExXprovides\overrightarrowand\overleftarrowcommands.+Some#additional>overandunderarrowcommandsareprovidedbytheamsmathpackqage#toUU llouttheset:7\overleftarrow.?\underleftarrow7\overrightarrow(\underrightarrow7\overleftrightarrow\underleftrightarrow~#3.4=1Dots#Whenktheamsmathpackqageisused,pellipsisdotsshouldnormallybGetypGedas#\dots. Placementݠ(onthebaselineorcentered)isdeterminedbywhatever#followsTthe\dots.IfthenextthingisaplussignorotherbinarysymbGol,#thedotswillbGecentered;ifit'sanyotherkindofsymbGol,ptheywillbeonthe#baseline. 2Ifϧthedotsfallattheendofamathformula,cthenextthingissomethinglike#\endπor\)or$, whichdoGesnotgiveanyinformationabGouthowtoplacethe#dots.fThen2+youmusthelpbyusing\dotscfor\dotswithcommas,"93or\dotsb#for)\dotswithbinaryopGerators/relations,"eor\dotsmfor\multiplicationdots,"#orUU\dotsifor\dotswithintegrals."qF*orexample,theinput~-Then?wehavetheseries$A_1,A_2,\dotsc$,-the?regionalsum$A_1+A_2+\dotsb$,-the?orthogonalproduct$A_1A_2\dotsm$,-and?theinfiniteintegral-\[\int_{A_1}\int_{A_2}\dotsi\]._@#will* proGducelowdotsinthe rstinstanceandcentereddotsintheothers,2with#theUUspacingoneithersideofthedotsnicelyadjusted:KThen(*wehavetheseriesA1|s;A2;:::,13the(*regionalsumA1Z+ފA2+<IUQ,UUtheorthogonalproGductA1|sA2'|j,andthein niteintegralcZ1Ɵ yA1GcZX yA2@W:m2SpGecifyingadotsthisway*,dinatermsoftheirmeaningratherthanintermsof#theirvisualplacement,isinkeepingwiththegeneralphilosophyofL5ffA͉TU>'ExXand#makesUUdoGcumentsmoreeasilyadaptabletodi erentconventions.7#3.5=1Non9breakingTdashes숍#Acommand\nobreakdashisprovidedtosuppressthepGossibilityofalinebreak#after<$thefollowinghyphenordash.&4F*orexample,uifyouwrite`pages1{9'as ՠs\ȍ8Y12&Z3. MISCELLANEOUSXMAJTHEMATICSFEATURES8GYpages?1\nobreakdash--9thenalinebreakwillneveroGccurbetweenthedash Yand#the9.2Y*oucanalsouse\nobreakdashtoprevent#undesirablehyphen-Yationsincombinationslike$p$-adic.bF*orfrequentuse,it'sadvisabletomakeYabbreviations,UUe.g.,#Y\newcommand{\p}{$p$\nobreakdash}%?for"\p-adic"Y\newcommand{\Ndash}{\nobreakdash--}%?for"pages1\Ndash9"Y%For?"\ndimensional"("n-dimensional"):Y\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}}BYThe/lastexampleshowshowtoprohibitalinebreakafterthehyphenbutallowYnormalhyphenationinthefollowingword.9(Itsucestoaddazero-widthspaceYafterUUthehyphen.)V;Y3.6s1Accen9tsTinmathՉYThe\followingaccentcommandsautomaticallygivegoGod\positioningofdoubleYaccents:c\Hat\Check \Tilde\Acute\Grave\Dot\Ddotc\Breve \Bar\VecBYWithstheusualnon-capitalizedmathaccentcommands,{ythesecondaccentwill9YsometimesmbGeaskew;forexample:U=^*x䍑{^A 뇲(\hat{\hat{A}}).$Withtheamsmathpack-Yage, eifyoutypGe\Hat{\Hat{A}}(usingthecapitalizedformforbothaccents)rYtheUUsecondaccentwillbGebetterpositioned:US^*x䍑X^qǵA Ȳ. hThisdoubleaccentopGerationiscomplicatedandtendstoslowdowntheYproGcessingofadocument.S]IfyourdoGcumentcontainsmanydoubleaccents, VyouYmaywishtousetheamsxtrapackqage,whichprovidesan\accentedsymbolYcommand.F\accentedsymbolQisasortofhybridof\newcommandand\savebox;YyouЕuseitinthepreambleofyourdoGcumenttostoretheresultofthedoubleYaccentUUcommandina`bGox'forquickretrievqal.c\accentedsymbol{\Ahathat}{\Hat{\Hat?A}}BhThecommands\dddotand\ddddotareavqailabletoproGducetripleandYquadruple-dotaccentsinadditiontothe\dotand\ddotaccentsalreadyavqail-YableUUinL5ffA͉TU>'ExX.V;Y3.7s1RoQotsՉYIn`ordinaryL5ffA͉TU>'ExXtheplacementofroGotindicesissometimesnotsogood:P zPpLzPfeVkY(\sqrt[\beta]{k}).In~theamsmathpackqage\leftrootand\uprootallowYyouUUtoadjustthepGositionoftheroot:c\sqrt[\leftroot{-2}\uproot{2}\beta]{k}BYwillMcmovethebGetaupandtotheright:QV( ܀zPp1ןzPfeVk-.o!ThenegativeargumentusedwithY\leftrootŲmovesthe ytotheright.JTheunitsareasmallamountthatisaYusefulUUsizeforsuchadjustments. s\ȍ8#3. MISCELLANEOUSXMAJTHEMATICSFEATURES&Z138G#3.8=1Bo9xedTformulasT#Theotcommand\boxedputsabGoxarounditsargument,ulike\fboxexceptthat #theUUcontentsareinmathmoGde:}ܟffmIff"C(`([ٲ)8+M\(0;))lO}ffffffmI#(3.2)Ѝ-\boxed{\eta?\leqC(\delta(\eta)+\Lambda_M(0,\delta))};#3.9=1ExtensibleTarro9ws#\xleftarrow̲and\xrightarrowproGducearrowsthatextendautomaticallyto#accommoGdate1unusuallywidesubscriptsorsuperscripts.]Thesecommandstake#onemoptionalargument(thesubscript)andonemandatoryargument(thesu-#pGerscript,UUpossiblyempty):AꍑTn+1$ >2(OBT΍ni12G `񍍑>!H`T%%C#(3.3)u-\xleftarrow{n+\mu-1}\quad?\xrightarrow[T]{n\pmi-1};#3.10BAxingTsym9bQolstoothersymbQols#L5ffA͉TU>'ExX..provides\stackrelforplacingasupGerscriptabove..abinaryrelation.dIn#the*Jamsmathpackqagetherearesomewhatmoregeneralcommands,_\overset#and\underset,#thatcanbGeusedtoplaceonesymbolaboveorbelowanother#symbGol,[whether'it'sarelationorsomethingelse.|=Theinput\overset{*}{X}ፑ#will[placeasupGerscript-sizeabove[theX:*S~XN;\undersetistheanalogfor#addingUUasymbGolunderneath. 2SeeUUalsothedescriptionof\sidesetinx7.3.-g#3.11BF ractionsTandrelatedconstructions#3.11.1KDisallo9wingTprimitiveT<8E3sXfractioncommands$#The8sixgeneralizedfractioncommands\over,q\overwithdelims,\atop,\atop-#withdelims,\above,\abovewithdelims areexpresslyforbiddenbytheams-#mathppackqage,astheirsyntaxisdecidedlyoutofplaceinL5ffA͉TU>'ExX;޼useoftheforms#\frac,UU\binom,\genfrac,andvqariantsisrequired.^2?#3.11.2KTheT\frac,\dfrac,and\tfraccommands#The\fraccommand,whichisinthebasiccommandsetofL5ffA͉TU>'ExX,takestwo#arguments|numeratorianddenominator|andtypGesetstheminnormalfraction# R\ffv J= "5-:2LNotonlyistheunÎusualsyntaxoftheprimitiveTAEMXfractioncommandsratheroutof placem&inL#A TAEMX,butfurthermorethatsynÎtaxseemstob'ExXbookishowtocontrolthesize#ofn9largedelimitersiftheautomaticsizingdoneby\leftand\rightproGduces#unsatisfactorysresults.Z'Theautomaticsizinghastwoslimitations:NVFirst,itisap-#pliedmechanicallytoproGducedelimiterslargeenoughtoencompassthelargest#containedzitem,andsecond,therangeofsizesisnotevenapproximatelycon-#tinuous4buthasfairlylargequantumjumps.fThismeansthatamathfragment#thatwisin nitesimallytoGolargeforagivendelimitersizewillgetthenextlarger#size,a jumpof3ptorsoinnormal-sizedtext.YJTherearetwo orthreesituations#wherethedelimitersizeiscommonlyadjusted,usingasetofcommandsthat#haveUU`big'intheirnames.0ٮ'-+Delimiter͟ ffb+text\left\bigl#\Bigl\bigglFC\Biggl-+ڲsize0` ffb+size\right\bigr#\Bigr\biggrFC\Biggr'+ٟffPN Result ff;(b)(<$c33wfe4r (֍dز)bs(fWUb)p06`<$w̙cw\Mwfe4r (֍d}`ןbR.bbWb<$ -c ]wfe4r (֍dņb`b``<$ Gc $wfe4r (֍d`ɟ^=b^\t^<$\gcwfe4r (֍dS^\ 'ȵb\!꬟\ <$x׵cwfe4r (֍dp0\!/5#The rstkindofsituationisacumulativeopGeratorwithlimitsaboveandbelow.#With&\leftand\rightthedelimitersusuallyturnoutlargerthannecessary*,Πs\ȍ8Y16&Z3. MISCELLANEOUSXMAJTHEMATICSFEATURES8GYandUUusingtheBigorbiggsizesinsteadgivesbGetterresults:%?ZY2 Y4X t^Ĵivai      TIX t شjpxij      (#ִpZ-'u3 -'u5ǟn*1=p'versusU^%'ExX\big,%\bigg,\Big,anda\Biggdelimitersaren'tscaledYpropGerly1overthefullrangeofL5ffA͉TU>'ExXfontsizes.JfWiththeamsmathpackqagetheyYare.UY3.14.2V erticalTbarnotationsYThewamsmathpackqageprovidescommands\lvert,M?\rvert,\lVert,\rVertY(compare!\langle,YS\rangle)toaddresstheproblemofoverloading!fortheYvertYbarcharacter|.KThischaracteriscurrentlyusedinL5ffA͉TU>'ExXdoGcumentstoYrepresentRawidevqarietyofmathematicalob8jects:the`divides'relationinaYnumbGer-theory;expressionlikepjq[ٲ,u%ortheabsolute-vqalueoperationjzpj,u%ortheY`suchthat'conditioninsetnotation,Korthe`evqaluatedat'notationfDz(t)  UUt=0.YThemultiplicityofusesinitselfisnotsobad; Zwhatisbad,however,isthatfactYthat"VnotalloftheusestakethesametypGographicaltreatment,UandthattheYcomplexediscriminatorypGowerseofaknowledgeablereadercannotbereplicatedYinMjcomputerproGcessingofmathematicaldocuments,Oatleastnotwithoutasig-Yni cant$costinproGcessingspeed,andeventhennotwithoutfallingsomewhatYshortͿofhumanreaders'abilities.DItisrecommendedthereforethatthereshouldYbGeMaone-to-onecorrespondenceinanygivendoGcumentbGetweenthevertbarYcharacter|andaselectedmathematicalnotation,andsimilarlyforthedouble-YbarBcommand\|.IlThisimmediatelyrulesouttheuseof|and\|fordelimiters,'s\ȍ8#4. OPERAJTORXNAMES(AMSOPN,AMSMATHPAÎCKAGES)yx178G#asinthenotationsforabsolutevqalueornorm,&bGecauseleftandrightdelimiters #aredistinctusagesthatdonotrelateinthesamewaytoadjacentsymbGols;&rec-#ommendedpracticeisthereforetode nesuitablecommandsinthedoGcument#preambleUUforanypaired-delimiteruseofvertbarsymbGols:{#\newcommand{\abs}[1]{\lvert#1\rvert}#\newcommand{\norm}[1]{\lVert#1\rVert}#whereupGoncbthedocumentwouldcontain\abs{z}toproGducejzpjand\norm{v}#toUUproGducekv[ٸk. #|4|#Op`eratornames(amsopn,amsmathpack@ages)#4.1=1De ningTnewopQeratornamesȍ#Mathfunctionssuchaslogۈ,sinL,andlimIaretraditionallytypGesetinromantypGe#toAmakethemvisuallymoredistinctfromone-lettermathvqariables,whichare#setAinmathitalic.kFThemorecommononeshaveAprede nednames,E\log,\sin,#\lim,20and)gsoforth,butnewonescomeupallthetimeinmathematicalpapGers,#sotheamsopnpackqageprovidesageneralmechanismforde ningnew`opGerator#names'.Astheamsopnpackqageisloadedinternallybytheamsmathpackqage,#thefollowingfeaturesareavqailabletherealso.TT*ode neamathfunction\xxx#toUUworklike\sin,youwrite#\DeclareMathOperator{\xxx}{xxx}#whereupGonoensuingusesof\xxxwillproducexxxintheproperfontandau-#tomaticallyaddpropGerspacingoneithersidewhennecessary*,sothatyouget#Axxx*Binstead>ofAxxxZBq.jInthesecondargumentof\DeclareMathOperator#(thenametext),7apseudo-textmoGdeprevqails::thehyphencharacter-willprint#asatexthyphenratherthanaminussignandanasterisk*willprintasaraised#text asteriskinsteadofacenteredmathstar.(Compareac-b*c\andabc.)#ButXotherwisethenametextisprintedinmathmoGde, Xsothatyoucanuse, Xe.g.,#subscriptsUUandsupGerscriptsthere.2If thenewopGeratorshouldhave subscriptsandsuperscriptsplacedin`limits'#pGosition(aboveandbGelowaswithlimع,]supkW,]ormax,usethe*formofthe#\DeclareMathOperatorUUcommand:#\DeclareMathOperator*{\Lim}{Lim}2ACfewC spGecialoperatornamesareprede nedbytheamsopnpackqage:h\var-#injlim,UU\varprojlim,\varliminf,and\varlimsup:+w\varlimsupf fe 㑟$limПn!1XQ(unq~;un^8u)0w\varliminfflimfljfe 㑎Пn!1jݵan+1jS=jqŵanq~j=0w\varinjlimflim3fJ'!J((mi>:)_0r\varprojlimflim3f rП:p2Sa(A)JApfj09s\ȍ8Y18Ve6. THEX\BOLDSYMBOLCOMMAND(AMSBSY,AMSMATHPJAÎCKAGES)8GY4.2s1\modTanditsrelativ9es΍YCommands\mod, \bmod,\pmod,\podareprovidedbytheamsopnpackqageto Ydeal5withthespGecialspacingconventions5of\mod"notation.g:\bmodand\pmodYareavqailableinL5ffA͉TU>'ExX,butwiththeamsopnpackqagethespacingof\pmodwillYadjusttoasmallervqalueifit'susedinanon-display-moGdeformula. e\modYand\podarevqariantsof\pmodpreferredbysomeauthors;Z\modomitstheYparentheses,UUwhereas\podomitsthe\moGd"andretainstheparentheses. }gcd(n;mmoGdߵn); xy [ɲ(moGdb);xy [ɲmoGd ȵc;xy [ɲ(d)Y(4.1)Y\gcd(n,m\bmod?n);\quadx\equivy\pmodbY;\quad?x\equivy\modc;\quadx\equivy\podd(Y|5|RYThe\textcommand(amstext,amsmathpack@ages)RYThe\textcommandisde nedbytheamsmathpackqagethroughasubGordi-YnateCpackqageamstext(whichcanalsobGeusedindependentlyifdesired).TheYmainuseofthecommand\textisforwordsorphrasesinadisplay*.fyItisYveryTbsimilartotheL5ffA͉TU>'ExXcommand\mboxinitse ects,%buthasacoupleofYadvqantages.ZAIfyouwantawordorphraseoftextinasubscript,2youcantypGeY..._{\text{word?orphrase}},4which,Yisslightlyeasierthanthe\mboxequiv-Yalent:-R..._{\mbox{\scriptsize3wordorphrase}}. TheotheradvqantageisYtheUUmoredescriptivename.!f:[xi1 Ǵ;xi*]"isUUmonotonic,gYi=1;:::;c8+1Y(5.1)Yf_{[x_{i-1},x_i]}?\text{ismonotonic,}Y\quad?i=1,\dots,c+1(Y|6|RYThe\boldsymbolcommand(amsbsy,amsmathpack@ages)RYThe\boldsymboland\pmbcommandsarede nedbytheamsbsypackqage(alsoYloaded;byamsmath).The\boldsymbolcommandisusedtoobtainbGoldnumbGersYandothernonalphabGeticsymbols,baswellasboldGreekletters,bwhichcannotYbGe7madeboldviathe\mathbfcommand.^1 "ItcanalsobeusedtoobtainboldYmathitalicletters;"XcomparetheresultsofM,\mathbf{M}and\boldsymbol{M}:YMM,DF cmmib10M .Y~ffv J= "5-:1LActuallyJ,dep'ExXonthedoGcumentationY leUUamsintx.dtxtogetthemostup-to-dateinformationonusage. ύY|8|ύYCommutativediagrams(amscdpack@age)ύYSomeQcommutativediagramcommandsliketheonesinAU>'M S-TU>'ExXareavqailableYasSaseparatepackqage,Tamscd.qAF*orcomplexcommutativediagramsauthorswillYneedtoturntomorecomprehensivepackqageslikeXY-UUpic(seexC.5),Lbutfor]6s\ȍ8#9. USINGXMAJTHFÎONTS218G#simple,diagramswithoutdiagonalarrowstheamscdcommandsmaybGemore #convenient.qHereUUisoneexample.*7"S^WLȸ 8Tꍒ̐7j$kÍ>UM!ŵT??y:6?:6?:6y8End wܴPLu(Sm 8Tc)=Ikffffffk(Z 8Tc)=J+#\begin{CD}#S^{{\mathcal{W}}_\Lambda}\otimes?T@>j>>T\\#@VVV@VV{\End?P}V\\#(S\otimes?T)/I^@=(Z\otimesT)/J#\end{CD}C#Inm=theCDm7environmentm=thecommands@>>>,s7@<<<,@VVV,m=and@AAAgiverespGec-#tivelyright,left,down,anduparrows.TF*orthehorizontalarrows,material#bGetweenGthe rstandsecond>or'ExX,1seetheL5ffATU>'ExXfont#guidey(fntguide.tex)orTheL雍Q1t}\cmti7ATKß'E-XComp}'anionyܲ[3].<~ManyyusersofAU>'M S-L5ffA͉TU>'ExX#alsoۗobtainanauxiliarycollectionofmathfontsknownas`AMSF*onts'.I3Theba-#sicsetofmathfontcommandsinL5ffA͉TU>'ExXincludes\mathbf,*x\mathrm,\mathcal,#\mathsf,f\mathtt,\mathit.)NAdditional{mathalphabGetcommandsareavqailable#throughthepackqagesamsfontsandeucal,riftherequisitefontsareinstalled#onUUyoursystem(seexC.2).;#9.2=1RecommendedTuseofmathfon9tcommands#If:you ndyourselfemployingmathfontcommandsfrequentlyinyourdoGcument,#youmightwishthattheyhadshorternames,suchas\mbinsteadof\mathbf.#OfQcourse,thereisnothingtokeepyoufromprovidingsuchabbreviationsfor#yourself4bysuitable\newcommandstatements.fButforL5ffA͉TU>'ExXtoprovideshorter#names16wouldactuallybGeadisservicetoauthors,8oasthatwouldobscureamuch#bGetter(galternative:[Pde ningcustomcommandnamesderivedfromthenamesof#the}underlyingmathematicalob8jects,ratherthanfromthenamesofthefonts#usedtodistinguishtheob8jects.9F*orexample,ifyouareusingbGoldtoindicate#vectors,sthenyouwillbGebetterservedinthelongrunifyoude nea`vector'#commandUUinsteadofa`math-bGold'command:gǠs\ȍ8Y22%10. 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THEOREMSXANDRELAJTEDSTRÎUCTURES8GhBydefaulteachkindoftheorem-likeenvironmentisnumbGeredindepen- Ydently*.ThusjAifyouhavethreelemmasandtwotheoremsinterspGersed,{theyYwillmbGenumberedmsomethinglikethis:PLemma1, Lemma2,Theorem1,LemmaY3,_Theorem]2.IfyouwantlemmasandtheoremstosharethesamenumbGeringYsequence|Lemma1,0NLemma2,Theorem3,Lemma4,Theorem5|thenyouYshouldUUindicatethedesiredrelationshipasfollows::ɍY\newtheorem{thm}{Theorem}Y\newtheorem{lem}[thm]{Lemma}YTheoptionalargument[thm]inthesecondstatementmeansthatthelemYenvironmentshouldsharethethmnumbGeringsequenceinsteadofhavingitsYownUUindepGendentsequence.hT*ojhaveatheorem-likeenvironmentnumbGeredsubordinatelywithinasec-Ytionalunit|e.g.,vtogetpropGositionsnumberedProposition2.1,vProposition2.2,YandmsooninSection2|putthenameoftheparentunitinsquarebracketsinY nalUUpGosition:Y\newtheorem{prop}{Proposition}[section]YWithvmtheoptionalargument[section],thepropcounterwillbGeresetto0YwheneverUUtheparentcountersectionisincremented.38Y10.4xChangingTst9ylesfortheorem-likeenvironmentsȍY10.4.1TheT\theoremstylecommandfdYTheuamsthmpackqagesuppGortsthenotionofacurrenttheoremstyle,}whichde-YterminesswhatwillbGeproducedbyagiven\newtheoremcommand.ͭThethreeYtheoremstylesprovided|plain,definition,andremark|receivedi erentty-YpGographicaltreatmentthatgivesthemvisualemphasiscorrespGondingtotheirYrelativeimpGortance.7ThedetailsofthistypGographicaltreatmentmayvqaryde-YpGendingBonthedocumentclass,~hbuttypicallytheplainstyleproGducesitalicYbGodyUUtext,whiletheothertwoUUstylesproGduceromanbodytext.hT*o7createnewtheorem-likeenvironmentsinthedi erentstyles,=divideyourY\newtheorem;commandsintogroupsandprefaceeachgroupwiththeappro-Ypriate`a\theoremstyle.Ifno\theoremstylecommandisgiven,c$thestyleusedYwillUUbGeplain.qSomeexamples:Y\theoremstyle{plain}%?defaultY\newtheorem{thm}{Theorem}[section]Y\newtheorem{lem}[thm]{Lemma}Y\newtheorem{prop}[thm]{Proposition}Y\newtheorem*{cor}{Corollary}Y\newtheorem*{KL}{Klein's?Lemma}Y\theoremstyle{definition}Y\newtheorem{defn}{Definition}[section]Y\newtheorem{conj}{Conjecture}[section]Y\newtheorem{exmp}{Example}[section]s\ȍ8#10.XTHEOREMSANDRELAJTEDSTRÎUCTURESv258G#\theoremstyle{remark} #\newtheorem*{rem}{Remark}#\newtheorem*{note}{Note}#\newtheorem{case}{Case}]#10.4.2KNum9bQerTswappingč#Anotuncommonstylevqariationfortheoremheadsistohavethetheorem#numbGer4ontheleft,latthebeginningoftheheading,linsteadofontheright.#As~ thisvqariationisusuallyappliedacrossthebGoardregardlessofindividual#\theoremstyleSchanges,SnumbGer-swappingisdonebyplacinga\swapnumbers#commandatthebGeginningofthelistof\newtheoremstatementsthatshould#bGeUUa ected.qF*orexample:]#\swapnumbers#\theoremstyle{plain}#\newtheorem{thm}{Theorem}#\theoremstyle{remark}#\newtheorem{rem}{Remark} #Afterx&theabGovex&declarations,theoremandremarkheadswillbeprintedinthe#formUU1.4TTheorem.,9.1.R}'emark._#10.4.3KF urtherTcustomizationpQossibilities#Moreajextensivecustomizationcapabilitiesareprovidedbytheamsthmpackqagein#the:formofa\newtheoremstylecommandandamechanismforusingpackqage#optionstoloadcustomtheoremstylede nitions.L0Asthesecapabilitiesaresome-#whatbGeyondtheneedsoftheaverageuser,discussionofthedetailsisconsigned#toUUtheexample lethmtest.texandtothecommentaryinamsthm.dtx.#10.5BProQofsd(#A.0prede ned.:proofenvironment.:providedbytheamsthmpackqageproGducesthe#heading>\ProGof"withappropriatespacingandpunctuation.-Theproofenvi-#ronment3isprimarilyintendedforshortproGofs,nomorethanapageortwoin#length;YlongerXproGofsareusuallybetterdoneasaseparate\sectionor\sub-#sectionUUinyourdoGcument. `2A`Q.E.D.'symbGol,offdffYffff ,isautomaticallyappendedattheendofaproof#environment.9T*osubstituteadi erentend-of-proGofsymbGol,Zuse\renewcommand#torede nethecommand\qedsymbol.]F*oralongproGofdoneasasubsectionor#sectionTinsteadofwiththeproofenvironment,T\youTcanobtainthesymbGoland#theUUusualamountofprecedingspacebyusing\qed.2PlacementkoftheQ.E.D.symbGolcanbeproblematicifthelastpartofa#proof]environmentisadisplayedequationorlistenvironmentorsomethingof#that={nature.iAdequateresultscansometimesbGeobtainedbyusing\qedatthe#appropriateuSspGotandthenunde ning\qedjustbeforetheendoftheproof.#(The3e ectwillbGeautomaticallylocalizedtothecurrentproofbynormalL5ffA͉TU>'ExX#scopingUUrules.)qF*orexample:s\ȍ8Y26%10. THEOREMSXANDRELAJTEDSTRÎUCTURES8GY\begin{proof} Y...Y\begin{equation}YG(t)=L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t)?\qedY\end{equation}Y\renewcommand{\qed}{}\end{proof}hAnToptionalargumentoftheproGofenvironmentallowsyoutosubstituteaYdi erent5nameforthestandard\ProGof".hIfyouwanttheproGofheadingtobe,Ysay*,UU\ProGofoftheMainTheorem",thenwriteY\begin{proof}[Proof?oftheMainTheorem]Ys\ȍ8#App'M S-L5ffA͉TU>'ExXitisnecessaryforyoutohavearecentversion #of.L5ffA͉TU>'ExX(June1994orlater,`L5ffATU>'ExX2"%S').SIfyou'renotsureabGouttheversion,#loGokb!atthestartupmessagethatisprintedonscreenandintheTU>'ExXlogwhen#youHrunL5ffA͉TU>'ExX.NItshouldmentiontheL5ffA͉TU>'ExXversionnumbGeranddatesomewhere#inthe rsttenlines.IfyourversionofL5ffA͉TU>'ExXisolderthanJune1994,Swe#suggestgettingthelatestversionfromtheComprehensiveTU>'ExXArchiveNetwork#(CT*AN),<:directory_tex-archive/macros/latex,A?ftpaddressesftp.cdrom.com#(US),;ftp.dante.de(Germany),@orftp.tex.ac.uk(UK).Ifftp letransferis#notfanoptionforyou,@jcontactfthesourcefromwhichyouoriginallyobtained#L5ffA͉TU>'ExX.#A.2@#PuttingT lesinasuitableplaceony9oursystem#SeeUUtheREAD.ME leforpGossibleupdatesaboutinstallationprocedures. 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S-L5ffA͉TU>'ExXma8jordoGc-Yumentstyles{ortheamsmathpackqage.P)Notallcategoriesarelisted;theonesthatYappGearUUaretheoneswhereproblemstendtooccurnowadays.hNoteinparticularthatthebasevqalueforstringpGoolneedstobGemuchlargerYthanthevqaluestypicallyfoundattheendofaL5ffA͉TU>'ExXlog.QBThisisbGecausetheYstringEpGoolcapacityreportedbyTU>'ExXinresponsetoa\tracingstatscommandYis,notthebasevqalue,abuttheresultofsubtractingfromthebasevaluetheYnumbGerofcharactersinTU>'ExX'sbuilt-inerrormessages,=thenamesofprimitiveYcontrolysequences,%andthenamesofalladditionalcontrolsequencesde nedinYthe6format le(inourcase,oathewholeofL5ffA͉TU>'ExX),nottomentionfontnamesYand# lenames.iThustherepGortedvqalueonlymeasurestheamountofstringYcapacityCthatremainstotheuseraftertheformat leisloaded.TherepGortedYvqalueUUfornumbGerUUofstringsisreducedinthesameway*./덍j^T ableTA.1.~.RecommendedUUvqaluesforselectedTU>'ExXmemorycategoriesR@qXL͉ff(N"fd͟ ff] ffCapacity!  ff( ffͤ ff/ fdCategory͡ ffcGAdequate͟ ffGenerous͟ 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