Table of Contents - University of the West Indies

Description

• cours - matière potentielle : outlines
• cours - matière potentielle : descriptions

• online business development unit
• bsc gen
• franklyn bsc chem engineering
• agricultural engineering ext
• uwi
• ext
• phd
• edu
• 4 edu
• bsc
• agriculture

Sujets

• Outline
• Hussein
• Bekele
• Gordon
• Temporary
• Sultán
• Eudoxie

Informations

• Publié par: urmu
• Nombre de lectures: 23
• Langue: English

Extrait

Abbreviations Patterns [DC, Tut] Mathematicacommands summary (cheat sheet) approx Approximation Blank: any expr arg Argument Hugo Touchette char Character Double blank: any seq of expr School of Mathematical Sciences, Queen Mary University of London cnt Counter Last updated: October 8, 2010 cond Condition Triple blank: zero or more args dist Distribution xAny expr namedx elem Element Equalities, assignments [DC] x hExpr namedxwith headh eqn Equation eval Evaluation xExact matchingx =Assign value expr Expression x+2/.x->2Output: 4 :=Also used for repeated evalDelayed eval fct Function x:pattNamed expr matchingpatt ==Equality patt Pattern f[aˆ b]/.f[x:ˆ ]->p[x]Output:p[aˆ b] rand Random =.Clear valueSame asClear str String x :vOptional arg with default valv ===Same asBoolean testSameQ val Value var Variable x h:vTyped arg with optional valv f[x ]:=f[x]=...Remember computed vals f[n ]fwith any named argn Rules and replacements [DC, Tut] f[n ,n ]fwith identical args xˆ nxto any named powern patt->exprTransformation ruleCan be assigned to names x ˆnAny expr to any power patt:->exprSame asDelayed rule:> a +bSum of two named exprs expr/.ruleReplacement SameasReplaceAll {a1 ,a2}List of two named exprs 1+2 x/.x->3Output: 7 patt/;condPattern with condRead:pattsuch thatcond expr/.{rule1,rule2}Multiple replacements (x /;NumberQ[x]&&Im[x]==0)Any real number expr//.ruleSame asReplace until no changesReplaceRepeated rule/;condRule with cond Applications, functions and maps [DC, Tut] patt?testPattern matching boolean test f[x ?NumberQ]:=x+2 f@exprPrefix formf[expr] (patt1|patt2|...) patt1orpatt2or ... expr//fPostfix formf[expr] {1,x,xˆ 2,yˆ 3})->q/.(x|xˆ nOutput:{1,q,q,yˆ 3} x˜f˜yInfix formf[x,y] expr/.ruleReplacement inexprwithrule f@@exprReplaces the head ofexprbyfSame asApply[f,expr] x+2/.x->2Output: 4 Plus@@{1,2}Output:1 + 2 = 3 f[a]+f[b]/.f[x ]->xˆ2Output:aˆ 2+bˆ 2 Similar to:f[x]/.f->gOutput:g[x] Position[f[a],g[b],f[b],f[x ]]Output:{{1},{2}} f/@listMapfonlistSame asMap[f,list] {1,x,xˆ 2,xˆ 3}->r[n]/.xˆ nOutput:{1,x,r[2],r[3]} f/@exprMapfon parts ofexpr Except[c]Anything exceptc Map[f,a+b]Output:f[a]+f[b] Except[c,patt]Patternpattexceptc Function[x,body]Function with argxSame as pure functions 2 Cases[{1,0,2},Except[0]]Output:{1,2} Function[x,xˆ 2][n]Output:n 2 2 Cases[{1,x,0},Except[0, Integer]]Output:{1} Map[Function[x,xˆ 2],a+b]Output:a+b Function[{vars},...]Many args Type specificationFiles [DC] body&Pure (anonymous) function IntegerN Esc -dsN- Esc Import["file"]Import file or url (#ˆ 2&)[5]Output: 25 RealR Esc -dsR- Esc 2 2 Export["file",expr]Exportexprin file Map[#ˆ 2&,a+b]Output:a+b ComplexC Esc -dsC- Esc ImageResolution->ptsImage resolution in pts (1+g[#])&/@{1,2}Output:{1+g[1],1+g[2]} RationalZ Esc -dsZ- Esc ReadList["file"]Returns list from file #nnth arg in pure function ListList type or head <<File‘Load content of file ##Seq of all args SymbolAny symbol StringString type of head Formatting [DC]Shortcuts [DC] √ HeadTypeHead Ctrl -@∙ FullFormDisplay full head form MatchQ[xˆ 2,Power] Ctrl -/Fraction MatrixFormDisplay in matrix form Ctrl -ˆSuperscript ColumnDisplay in column form Ctrl -Subscript TableFormDisplay in tabular form Ctrl - Enter New row TreeFormDisplay in tree structure Ctrl -,New column

Lists [DC, Tut] {i,imax} {i,imin,imax,di} Table[expr,cnt] list[[i]] list[[i,j]] list[[i;;j]] Drop[list,n] Drop[list,{n}] Extract[expr,list] Array[f,n]

Range[n] Range[nmin,nmax,dn] Position[expr,patt] Count[list,patt] Min[list], Max[list] Append[expr,elem] AppendTo[expr,elem] Prepend[expr,elem] Insert[list,elem,n] Delete[expr,elem,n] ReplacePart[expr,i->new] ReplaceList[expr,rule] Flatten[list] Flatten[list,n] Take[list,n]

Cases[list,patt]

Cases[list,patt,n] Cases[list,rule] Select[list,boolean] Position[expr,patt] Join[l1,l2,...] Union[l1,l2,...] Riffle[{e1,e2,...},x] Intersection[l1,l2,...] Subsets[list] DeleteDuplicates Thread[f[{a,b,c}]] Sort[list] Reverse[list] PadLeft[list,n,x] PadRight[list,n,x] Partition[list,n] Split[list] Tuples[list,n] Permutations[list] Outer[f,l1,l2]

Boolean functions [DC]Programming [DC, Tut1, Tut2] Simple counter from 1 toimax SameQ[x,y] Same? (x===y) Do[expr,cnt]Do loop Counter with step list di OrderedQ[list] ordered? While[test,body]While loop Generate listMultiple counters possible IntegerQ[x]xinteger? If[cond,t,f]If statement iSame asth elem (list or expr)Part[list,i]orlist[i] MemberQ[list,patt]Elems matchpatt? Return[expr]Outputexprand stop (i, j)elem Sameaslist[[i]][[j]] MatchQ[expr,patt] exprmatchespatt? Module[{x,...},expr]Proc with local vars RangeitojSame asPart[list,i;;j] ValueQ[expr] exprhas value? (vars localization) FirstnLast ifelems dropped-n AtomQ[expr] expris atomic? Block[{x,...},expr]Eval with global vars nth elem dropped FreeQ[expr,patt] exprfree ofpatt? With[{x=x0,...},expr]Local constants Extract inexprat poslist EvenQ[n] neven integer? Trace[expr]Trace eval ofexpr {f(1), f(2), . . . , f(n)} OddQ[n]nodd integer? Compile[{vars},expr]Compiled expr Array[1+#ˆ 2&,2]Output:{2,5} PrimeQ[n]nprime integer? i++Incrementation {1,2,3, . . . , n}Same as expr Array[#&,n] NumberQ[expr] a number? i--Decrementation Full syntax NumericQ[expr] exprnumerical? i+=diIncrement bydi List matching positions PolynomialQ[expr,var] expra polynomial? i-=diDecrement bydi Number of matches VectorQ[expr] expr1D array (list)? i*=cMultiply bycand override val Min, max of listSame as: expr Min[i,j,...] MatrixQ[expr] 2D array (list)? i/=cDivide bycand override val exprwithelemadded at end DigitQ[str]All digits instr? Append and updateexprSame asexpr=Append[expr,elem] LetterQ[str] All letters instr? Inset at beginningSee also:PrependTo Inserteleminlistat posn Strings [DC, Tut] Deleteeleminlistat posn "chars"String Replacesexpr[[i]]bynew ToString[expr]Transformsexprto string Replace according torule StringJoin[str1,str2,...]Join stringsSame as<> Flattens nested lists StringLength[str]Number of chars instr Flattens to leveln StringTake[str,n]nfirst chars ofstrLast if-n Take firstnelem StringDrop[str,n]Firstnchars droppedLast if-n Take[#,1]&/@{{1,2},{3,1}}Output:{{1},{3}} StringInsert[str1,str2,n]Insertstr2at posn Elems oflistmatchingpatt 2 StringPosition[str,substr]Start and end ofsubstrinstr Cases[{2,xˆ 2},xˆ n]Output:x StringCount[str,substr]Number of occurrences ofsubstr Firstnmatchingpatt StringReplace[str,rule]Replace with rule Results for whichruleapplies StringMatchQ[str,patt] pattappears instr? Boolean selection inlist StringFreeQ[str,patt] strfree ofpatt? Pos wherepattappears StringSplit[str]Split where whitespace Join lists StringSplit[str,patt]Split according topatt Set unionSame as∪ SortAlphabetical (canonical) sort {e1,x,e2,x,...} DateString["elem"]String version of date Set intersectionSame as∩ Characters[str]List of chars All subsets oflist ToUpperCase[str]Capitalize Delete duplicates from list ToLowerCase[str]Lower case {f[a],f[b],f[c]} CharacterRange["char1","char2"]List fromchar1tochar2 Sortslist patt.. pattfollowed by symbols listin reverse order WhitespaceWhite space Addxntimes left oflistPad with 0 if nox Addxntimes right oflist Dynamic objects [ DC, Tut2 ] Partitionlistinnsublists Dynamic[expr]Dynamically updated expr Splitslistin identical elems DynamicModule[{vars},...]Module with dynamical vars n-tuples oflist Manipulate[expr,{u,...}]Interactive widget Permutations Outer (cartesian) product

Useful functions Sum[f,{i,imin,imax,di}] Product[f,{i,imin,imax,di}] Mod[m,n] Solve[f[x]==0,x] Expand[p] Factor[p] Length[expr] Numerator[expr] Denominator[expr] Options[cmd] SetOptions[cmd] Evaluate[expr] Piecewise[{{val,test},...}] InverseFunction[f][x] Nest[f,x,n] NestList[f,x,n] FixedPoint[f,x] FixedPointList[f,x] NestWhile[f,x,test] NestWhileList LinearModelFit NonlinearModelFit GraphPlot[m] GraphData["name"] FinancialData["tag","date"] CountryData["tag","prop"] WeatherData["tag","prop"]

Sum Product Remainder ofm/n Solves polynomial eqn Expand polynomialp Factorizep Number of elems inexpr Numerator ofexpr Denominator ofexpr Display options forcmd Set options forcmd Explicit eval Piecewise definition Inverse fct n-fold fct composition List of compositions Iterate until fixed point Iteration list While iteration While iteration list Linear model Nonlinear model Graph (network) for adjacencym Graph withname Financial data fromdate Country data Weather data

Probability and statistics [DC] RandomReal[{a,b}]Uniform rand real in[a, b] RandomReal[dist]Rand real fromdist RandomInteger[{i,j}]Uniform integer in[i, j] RandomInteger[dist]Rand integer fromdist RandomSample[list,n]nsamples fromlist RandomSeed[]Reset seed PDF[dist,x]Density fct ofdist CDF[dist,x]Cumulant density fct ofdist Histogram[list,w]Histogram (bin widthw)

BinCounts[list,w]Counting dist oflist(bin widthw) Mean[list]AlsoMean[dist] Variance[list] StandardDeviation[list] ChiSquareDistribution[ν] ExponentialDistribution[λ] NormalDistribution[µ,σ] BinomialDIstribution[n,p] BernoulliDistribution[p]

From[0,1]if no arg

Uniform0or1if no arg

Option:"Probability" "ProbabilityDensity"

Linear algebra [DC] a.b Cross[a,b] Norm[a] IdentityMatrix[n] Diagonal[m] Diagonal[m,k] Dimensions[m] Inverse[m] Det[m] Tr[m] Transpose[l] Eigenvalues[m] Eigenvectors

Parallel computation [DC, Tut] Dot or matrix product Parallelize[expr]Direct parallelization Cross product ParallelEvaluate[expr]Eval on kernels Euclidean norm DistributeDefinitionsPut def on kernels n×nidentity matrix ParallelNeedsPackage on kernels Diagonal ofm ParallelMapParallelMap kth elem in diagonal ofm ParallelSumParallelSum Dimensions ofm ParallelProductParallelProduct Inverse square matrixm ParallelTableParallelTable Determinant square matrix ParallelDoParallelDo Trace matrix or tensorm ParallelSubmitSubmit to kernel Transpose first two levels Collect withWaitAll Eigenvals square matrix WaitAllResults from kernels Eigenvects square matrix

Numerical routines [DC, Tut] N[expr]Numerical eval ofexpr NSolve[eqn,var]Approx polynomial eqn FindRoot[eqn,{x,x0}]Approx eqn in varxwith seedx0 FindRoot[eqn,{x,x0,a,b}]Approxeqnin[a, b] NDSolve[eqn,y,{x,a,b}]Solve differentialeqn NIntegrate[f,{x,a,b}]Numerical integration. Methods: "SymbolicProcessing"->0 "GlobalAdaptive" "LocalAdaptive" "Oscillatory" "PrincipalValue" "MonteCarlo" "AdaptiveMonteCarlo" FindMinimum[f,x,x0]Find local min with seedx0 NMinimize[f,x]Find global min. Methods: "DifferentialEvolution" "NelderMead" "RandomSearch" "SimulatedAnnealing" NumericalCalculuspackage: NLimit[expr,x->x0]Numerical limit ND[expr,x,x0]NumericalDatx0 NSeries[f,{x,x0,n}]Numerical series Notes

Initial cond specified ineqn

Fully numerical eval Default

1D integrals; automatic detection Cauchy principal value Random sampling

Option:"SearchPoints"

Option: Option:

"SearchPoints" "SearchPoints"

CopyrightHugo Touchette c 2010 Available at:http://www.maths.qmul.ac.uk/˜ht Mathematica is a trademark of Wolfram Research