├── .gitignore ├── 00-01-AdHoc.tex ├── 00-01-AttachmentToTextbookProjectsOnPlanetMath.tex ├── 00-01-Dimension.tex ├── 00-01-Expression.tex ├── 00-01-FusionOfTheoreticalPhysicsWithMathematicsAtIHESOrg.tex ├── 00-01-InfixNotation.tex ├── 00-01-Rigid.tex ├── 00-01-Strict.tex ├── 00-01-TextbookProjectsOnPlanetMath.tex ├── 00-01-ToyTheorem.tex ├── 00-02-AMSMSCClassificationOfArticlesAndConversionTables.tex ├── 00-02-FailureFunctionTests.tex ├── 00-02-GeorgePolya.tex ├── 00-02-MSCClassificationOfObjectsArticlesSearch.tex ├── 00-02-PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers.tex ├── 00A05-AnExampleOfMathematicalInduction.tex ├── 00A05-ArgMinAndArgMax.tex ├── 00A05-Arithmetic.tex ├── 00A05-ArithmeticProgression.tex ├── 00A05-ArithmeticgeometricharmonicMeansInequality.tex ├── 00A05-Arity.tex ├── 00A05-BabylonianMethodOfComputingSquareRoots.tex ├── 00A05-BiogroupoidsMathematicalModelsOfSpeciesEvolution.tex ├── 00A05-Calculator.tex ├── 00A05-Characterisation.tex ├── 00A05-ConwaysChainedArrowNotation.tex ├── 00A05-DecimalPlace.tex ├── 00A05-DecimalPoint.tex ├── 00A05-Definition.tex ├── 00A05-Division.tex ├── 00A05-FundamentalTheoremsInMathematics.tex ├── 00A05-GrahamsNumber.tex ├── 00A05-Grapher.tex ├── 00A05-Hypothesis.tex ├── 00A05-Indeterminate.tex ├── 00A05-IntegralSign.tex ├── 00A05-Introducing0thPower.tex ├── 00A05-KnuthsUpArrowNotation.tex ├── 00A05-Lemma.tex ├── 00A05-MacOSCalculator.tex ├── 00A05-MetricSystem.tex ├── 00A05-MockupOfABasicCalculator.tex ├── 00A05-MockupOfScientificCalculator.tex ├── 00A05-NegativeNumber.tex ├── 00A05-Parametre.tex ├── 00A05-Percentage.tex ├── 00A05-Proof.tex ├── 00A05-Property.tex ├── 00A05-RydiasMathemagicMinute.tex ├── 00A05-SaddlePointApproximation.tex ├── 00A05-Singleton.tex ├── 00A05-StableSubspace.tex ├── 00A05-Subsequence.tex ├── 00A05-SumOfOddNumbers.tex ├── 00A05-Supercomputers.tex ├── 00A05-SurrealNumber.tex ├── 00A05-TimeInvariant.tex ├── 00A05-Variable.tex ├── 00A05-VennDiagram.tex ├── 00A05-Welldefined.tex ├── 00A05-WindowsCalculator.tex ├── 00A06-DoublingAndHalvingAlgorithmForIntegerMultiplication.tex ├── 00A06-Multiplication.tex ├── 00A06-PlusSign.tex ├── 00A06-SimpleInterest.tex ├── 00A06-Subtraction.tex ├── 00A06-TableOfDivisionUpTo12.tex ├── 00A06-TableOfMultiplicationUpTo12.tex ├── 00A06-ThereforeSign.tex ├── 00A07-NesbittsInequality.tex ├── 00A07-ProofOfNesbittsInequality.tex ├── 00A08-CabtaxiNumber.tex ├── 00A08-IndexOfImportantIrrationalConstants.tex ├── 00A08-LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind.tex ├── 00A08-SmallIntegersThatAreOrMightBeTheLargestOfTheirKind.tex ├── 00A08-TaxicabNumbers.tex ├── 00A08-TopTenCoolestNumbers.tex ├── 00A20-Canonical.tex ├── 00A20-CompletingTheSquare.tex ├── 00A20-Counterexample.tex ├── 00A20-DigitalObject.tex ├── 00A20-FamousOpenQuestionsInMathematics.tex ├── 00A20-HighSchoolMathematics.tex ├── 00A20-IndexOfTables.tex ├── 00A20-MathematicsVocabulary.tex ├── 00A20-Obvious.tex ├── 00A20-OnLineEncyclopediaOfIntegerSequences.tex ├── 00A20-OpenQuestion.tex ├── 00A20-OverviewOfTheContentOfPlanetMath.tex ├── 00A20-Pathological.tex ├── 00A20-QuotationsAboutMathematics.tex ├── 00A20-TermsFromForeignLanguagesUsedInMathematicspageImagesVersion.tex ├── 00A20-TopicEntryOnAlgebra.tex ├── 00A20-Vacuous.tex ├── 00A30-ComplexSystemsBiologyCSB.tex ├── 00A30-MatheRealism.tex ├── 00A30-UltracomplexSystems.tex ├── 00A35-ErrorsCanCancelEachOtherOut.tex ├── 00A35-Needleinthehaystack.tex ├── 00A35-PreservationAndReflection.tex ├── 00A35-ProbabilisticProof.tex ├── 00A35-TechniquesInMathematicalProofs.tex ├── 00A69-CompoundInterest.tex ├── 00A69-Interest.tex ├── 00A69-MathematicsOfFinance.tex ├── 00A99-Arrow.tex ├── 00A99-BlackboardBold.tex ├── 00A99-EmptyProduct.tex ├── 00A99-JordansInequality.tex ├── 00A99-KochelNumber.tex ├── 00A99-LaTeXSymbolForCauchyPrincipalValue.tex ├── 00A99-OpusNumber.tex ├── 00A99-OrderOfOperations.tex ├── 00A99-PlanetMathFontSandbox.tex ├── 00A99-QED.tex ├── 00A99-QHoldingBay47.tex ├── 00A99-SevensegmentDisplay.tex ├── 00A99-SiegelsAxiom.tex ├── 00A99-Superscript.tex ├── 00A99-TFAE.tex ├── 00A99-TermsFromForeignLanguagesUsedInMathematicshtmlVersion.tex ├── 00A99-TextFigures.tex ├── 00A99-TopicEntryOnMiscellaneousMathematics.tex ├── 00A99-TriceSequence.tex ├── 00A99-Triskaidekaphobia.tex ├── 00A99-WLOG.tex ├── 00A99-WordsForNumbersInSlavicLanguages.tex ├── 00B60-TopicOnMajorResourcesForMathematicsAndMathematicalPhysics.tex ├── LICENSE ├── README.md ├── html ├── 00-01-AdHoc.html ├── 00-01-AttachmentToTextbookProjectsOnPlanetMath.html ├── 00-01-Dimension.html ├── 00-01-Expression.html ├── 00-01-FusionOfTheoreticalPhysicsWithMathematicsAtIHESOrg.html ├── 00-01-InfixNotation.html ├── 00-01-Rigid.html ├── 00-01-Strict.html ├── 00-01-TextbookProjectsOnPlanetMath.html ├── 00-01-ToyTheorem.html ├── 00-02-AMSMSCClassificationOfArticlesAndConversionTables.html ├── 00-02-FailureFunctionTests.html ├── 00-02-GeorgePolya.html ├── 00-02-MSCClassificationOfObjectsArticlesSearch.html ├── 00-02-PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers.html ├── 00A05-AnExampleOfMathematicalInduction.html ├── 00A05-ArgMinAndArgMax.html ├── 00A05-Arithmetic.html ├── 00A05-ArithmeticProgression.html ├── 00A05-ArithmeticgeometricharmonicMeansInequality.html ├── 00A05-Arity.html ├── 00A05-BabylonianMethodOfComputingSquareRoots.html ├── 00A05-BiogroupoidsMathematicalModelsOfSpeciesEvolution.html ├── 00A05-Calculator.html ├── 00A05-Characterisation.html ├── 00A05-ConwaysChainedArrowNotation.html ├── 00A05-DecimalPlace.html ├── 00A05-DecimalPoint.html ├── 00A05-Definition.html ├── 00A05-Division.html ├── 00A05-FundamentalTheoremsInMathematics.html ├── 00A05-GrahamsNumber.html ├── 00A05-Grapher.html ├── 00A05-Hypothesis.html ├── 00A05-Indeterminate.html ├── 00A05-IntegralSign.html ├── 00A05-Introducing0thPower.html ├── 00A05-KnuthsUpArrowNotation.html ├── 00A05-Lemma.html ├── 00A05-MacOSCalculator.html ├── 00A05-MetricSystem.html ├── 00A05-MockupOfABasicCalculator.html ├── 00A05-MockupOfScientificCalculator.html ├── 00A05-NegativeNumber.html ├── 00A05-Parametre.html ├── 00A05-Percentage.html ├── 00A05-Proof.html ├── 00A05-Property.html ├── 00A05-RydiasMathemagicMinute.html ├── 00A05-SaddlePointApproximation.html ├── 00A05-Singleton.html ├── 00A05-StableSubspace.html ├── 00A05-Subsequence.html ├── 00A05-SumOfOddNumbers.html ├── 00A05-Supercomputers.html ├── 00A05-SurrealNumber.html ├── 00A05-TimeInvariant.html ├── 00A05-Variable.html ├── 00A05-VennDiagram.html ├── 00A05-Welldefined.html ├── 00A05-WindowsCalculator.html ├── 00A06-DoublingAndHalvingAlgorithmForIntegerMultiplication.html ├── 00A06-Multiplication.html ├── 00A06-PlusSign.html ├── 00A06-SimpleInterest.html ├── 00A06-Subtraction.html ├── 00A06-TableOfDivisionUpTo12.html ├── 00A06-TableOfMultiplicationUpTo12.html ├── 00A06-ThereforeSign.html ├── 00A07-NesbittsInequality.html ├── 00A07-ProofOfNesbittsInequality.html ├── 00A08-CabtaxiNumber.html ├── 00A08-IndexOfImportantIrrationalConstants.html ├── 00A08-LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind.html ├── 00A08-SmallIntegersThatAreOrMightBeTheLargestOfTheirKind.html ├── 00A08-TaxicabNumbers.html ├── 00A08-TopTenCoolestNumbers.html ├── 00A20-Canonical.html ├── 00A20-CompletingTheSquare.html ├── 00A20-Counterexample.html ├── 00A20-DigitalObject.html ├── 00A20-FamousOpenQuestionsInMathematics.html ├── 00A20-HighSchoolMathematics.html ├── 00A20-IndexOfTables.html ├── 00A20-MathematicsVocabulary.html ├── 00A20-Obvious.html ├── 00A20-OnLineEncyclopediaOfIntegerSequences.html ├── 00A20-OpenQuestion.html ├── 00A20-OverviewOfTheContentOfPlanetMath.html ├── 00A20-Pathological.html ├── 00A20-QuotationsAboutMathematics.html ├── 00A20-TermsFromForeignLanguagesUsedInMathematicspageImagesVersion.html ├── 00A20-TopicEntryOnAlgebra.html ├── 00A20-Vacuous.html ├── 00A30-ComplexSystemsBiologyCSB.html ├── 00A30-MatheRealism.html ├── 00A30-UltracomplexSystems.html ├── 00A35-ErrorsCanCancelEachOtherOut.html ├── 00A35-Needleinthehaystack.html ├── 00A35-PreservationAndReflection.html ├── 00A35-ProbabilisticProof.html ├── 00A35-TechniquesInMathematicalProofs.html ├── 00A69-Interest.html ├── 00A69-MathematicsOfFinance.html ├── 00A99-Arrow.html ├── 00A99-BlackboardBold.html ├── 00A99-EmptyProduct.html ├── 00A99-JordansInequality.html ├── 00A99-KochelNumber.html ├── 00A99-LaTeXSymbolForCauchyPrincipalValue.html ├── 00A99-OpusNumber.html ├── 00A99-OrderOfOperations.html ├── 00A99-PlanetMathFontSandbox.html ├── 00A99-QED.html ├── 00A99-QHoldingBay47.html ├── 00A99-SevensegmentDisplay.html ├── 00A99-SiegelsAxiom.html ├── 00A99-Superscript.html ├── 00A99-TFAE.html ├── 00A99-TermsFromForeignLanguagesUsedInMathematicshtmlVersion.html ├── 00A99-TextFigures.html ├── 00A99-TopicEntryOnMiscellaneousMathematics.html ├── 00A99-TriceSequence.html ├── 00A99-Triskaidekaphobia.html ├── 00A99-WLOG.html ├── 00A99-WordsForNumbersInSlavicLanguages.html └── 00B60-TopicOnMajorResourcesForMathematicsAndMathematicalPhysics.html └── pdf ├── 00-02-GeorgePolya.pdf ├── 00A05-AnExampleOfMathematicalInduction.pdf ├── 00A05-ArgMinAndArgMax.pdf ├── 00A05-Arithmetic.pdf ├── 00A05-ArithmeticProgression.pdf ├── 00A05-ArithmeticgeometricharmonicMeansInequality.pdf ├── 00A05-Arity.pdf ├── 00A05-BabylonianMethodOfComputingSquareRoots.pdf ├── 00A05-Calculator.pdf ├── 00A05-Characterisation.pdf ├── 00A05-ConwaysChainedArrowNotation.pdf ├── 00A05-DecimalPoint.pdf ├── 00A05-Definition.pdf ├── 00A05-Division.pdf ├── 00A05-FundamentalTheoremsInMathematics.pdf ├── 00A05-GrahamsNumber.pdf ├── 00A05-Hypothesis.pdf ├── 00A05-Indeterminate.pdf ├── 00A05-IntegralSign.pdf ├── 00A05-Introducing0thPower.pdf ├── 00A05-KnuthsUpArrowNotation.pdf ├── 00A05-Lemma.pdf ├── 00A05-MacOSCalculator.pdf ├── 00A05-MetricSystem.pdf ├── 00A05-MockupOfABasicCalculator.pdf ├── 00A05-MockupOfScientificCalculator.pdf ├── 00A05-Parametre.pdf ├── 00A05-Percentage.pdf ├── 00A05-Proof.pdf ├── 00A05-Property.pdf ├── 00A05-RydiasMathemagicMinute.pdf ├── 00A05-SaddlePointApproximation.pdf ├── 00A05-Singleton.pdf ├── 00A05-StableSubspace.pdf ├── 00A05-Subsequence.pdf ├── 00A05-SumOfOddNumbers.pdf ├── 00A05-SurrealNumber.pdf ├── 00A05-TimeInvariant.pdf ├── 00A05-Variable.pdf ├── 00A05-Welldefined.pdf ├── 00A05-WindowsCalculator.pdf ├── 00A06-DoublingAndHalvingAlgorithmForIntegerMultiplication.pdf ├── 00A06-Multiplication.pdf ├── 00A06-PlusSign.pdf ├── 00A06-SimpleInterest.pdf ├── 00A06-Subtraction.pdf ├── 00A06-TableOfDivisionUpTo12.pdf ├── 00A06-TableOfMultiplicationUpTo12.pdf ├── 00A06-ThereforeSign.pdf ├── 00A07-NesbittsInequality.pdf ├── 00A07-ProofOfNesbittsInequality.pdf ├── 00A08-CabtaxiNumber.pdf ├── 00A08-IndexOfImportantIrrationalConstants.pdf ├── 00A08-LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind.pdf ├── 00A08-SmallIntegersThatAreOrMightBeTheLargestOfTheirKind.pdf ├── 00A08-TaxicabNumbers.pdf ├── 00A08-TopTenCoolestNumbers.pdf ├── 00A20-Canonical.pdf ├── 00A20-CompletingTheSquare.pdf ├── 00A20-Counterexample.pdf ├── 00A20-DigitalObject.pdf ├── 00A20-HighSchoolMathematics.pdf ├── 00A20-IndexOfTables.pdf ├── 00A20-MathematicsVocabulary.pdf ├── 00A20-Obvious.pdf ├── 00A20-OnLineEncyclopediaOfIntegerSequences.pdf ├── 00A20-OpenQuestion.pdf ├── 00A20-OverviewOfTheContentOfPlanetMath.pdf ├── 00A20-Pathological.pdf ├── 00A20-QuotationsAboutMathematics.pdf ├── 00A20-TermsFromForeignLanguagesUsedInMathematicspageImagesVersion.pdf ├── 00A20-TopicEntryOnAlgebra.pdf ├── 00A35-ErrorsCanCancelEachOtherOut.pdf ├── 00A35-Needleinthehaystack.pdf ├── 00A35-PreservationAndReflection.pdf ├── 00A35-ProbabilisticProof.pdf ├── 00A35-TechniquesInMathematicalProofs.pdf ├── 00A69-CompoundInterest.pdf ├── 00A69-Interest.pdf ├── 00A69-MathematicsOfFinance.pdf ├── 00A99-Arrow.pdf ├── 00A99-BlackboardBold.pdf ├── 00A99-EmptyProduct.pdf ├── 00A99-JordansInequality.pdf ├── 00A99-KochelNumber.pdf ├── 00A99-LaTeXSymbolForCauchyPrincipalValue.pdf ├── 00A99-OpusNumber.pdf ├── 00A99-OrderOfOperations.pdf ├── 00A99-PlanetMathFontSandbox.pdf ├── 00A99-QED.pdf ├── 00A99-QHoldingBay47.pdf ├── 00A99-SiegelsAxiom.pdf ├── 00A99-Superscript.pdf ├── 00A99-TFAE.pdf ├── 00A99-TermsFromForeignLanguagesUsedInMathematicshtmlVersion.pdf ├── 00A99-TopicEntryOnMiscellaneousMathematics.pdf ├── 00A99-TriceSequence.pdf ├── 00A99-Triskaidekaphobia.pdf ├── 00A99-WLOG.pdf ├── 00A99-WordsForNumbersInSlavicLanguages.pdf └── nocando /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | 11 | ## Intermediate documents: 12 | *.dvi 13 | *-converted-to.* 14 | # these rules might exclude image files for figures etc. 15 | # *.ps 16 | # *.eps 17 | # *.pdf 18 | 19 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 20 | *.bbl 21 | *.bcf 22 | *.blg 23 | *-blx.aux 24 | *-blx.bib 25 | *.brf 26 | *.run.xml 27 | 28 | ## Build tool auxiliary files: 29 | *.fdb_latexmk 30 | *.synctex 31 | *.synctex.gz 32 | *.synctex.gz(busy) 33 | *.pdfsync 34 | 35 | ## Auxiliary and intermediate files from other packages: 36 | # algorithms 37 | *.alg 38 | *.loa 39 | 40 | # achemso 41 | acs-*.bib 42 | 43 | # amsthm 44 | *.thm 45 | 46 | # beamer 47 | *.nav 48 | *.snm 49 | *.vrb 50 | 51 | # cprotect 52 | *.cpt 53 | 54 | #(e)ledmac/(e)ledpar 55 | *.end 56 | *.[1-9] 57 | *.[1-9][0-9] 58 | *.[1-9][0-9][0-9] 59 | *.[1-9]R 60 | *.[1-9][0-9]R 61 | *.[1-9][0-9][0-9]R 62 | *.eledsec[1-9] 63 | *.eledsec[1-9]R 64 | *.eledsec[1-9][0-9] 65 | *.eledsec[1-9][0-9]R 66 | *.eledsec[1-9][0-9][0-9] 67 | *.eledsec[1-9][0-9][0-9]R 68 | 69 | # glossaries 70 | *.acn 71 | *.acr 72 | *.glg 73 | *.glo 74 | *.gls 75 | 76 | # gnuplottex 77 | *-gnuplottex-* 78 | 79 | # hyperref 80 | *.brf 81 | 82 | # knitr 83 | *-concordance.tex 84 | *.tikz 85 | *-tikzDictionary 86 | 87 | # listings 88 | *.lol 89 | 90 | # makeidx 91 | *.idx 92 | *.ilg 93 | *.ind 94 | *.ist 95 | 96 | # minitoc 97 | *.maf 98 | *.mtc 99 | *.mtc[0-9] 100 | *.mtc[1-9][0-9] 101 | 102 | # minted 103 | _minted* 104 | *.pyg 105 | 106 | # morewrites 107 | *.mw 108 | 109 | # mylatexformat 110 | *.fmt 111 | 112 | # nomencl 113 | *.nlo 114 | 115 | # sagetex 116 | *.sagetex.sage 117 | *.sagetex.py 118 | *.sagetex.scmd 119 | 120 | # sympy 121 | *.sout 122 | *.sympy 123 | sympy-plots-for-*.tex/ 124 | 125 | # pdfcomment 126 | *.upa 127 | *.upb 128 | 129 | #pythontex 130 | *.pytxcode 131 | pythontex-files-*/ 132 | 133 | # Texpad 134 | .texpadtmp 135 | 136 | # TikZ & PGF 137 | *.dpth 138 | *.md5 139 | *.auxlock 140 | 141 | # todonotes 142 | *.tdo 143 | 144 | # xindy 145 | *.xdy 146 | 147 | # xypic precompiled matrices 148 | *.xyc 149 | 150 | # WinEdt 151 | *.bak 152 | *.sav 153 | 154 | # endfloat 155 | *.ttt 156 | *.fff 157 | 158 | # Latexian 159 | TSWLatexianTemp* 160 | -------------------------------------------------------------------------------- /00-01-AdHoc.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AdHoc} 4 | \pmcreated{2013-03-22 14:45:20} 5 | \pmmodified{2013-03-22 14:45:20} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{ad hoc} 9 | \pmrecord{9}{36396} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{amsthm} 27 | 28 | \usepackage{mathrsfs} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | % 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \newcommand{\sR}[0]{\mathbb{R}} 44 | \newcommand{\sC}[0]{\mathbb{C}} 45 | \newcommand{\sN}[0]{\mathbb{N}} 46 | \newcommand{\sZ}[0]{\mathbb{Z}} 47 | 48 | \usepackage{bbm} 49 | \newcommand{\Z}{\mathbbmss{Z}} 50 | \newcommand{\C}{\mathbbmss{C}} 51 | \newcommand{\R}{\mathbbmss{R}} 52 | \newcommand{\Q}{\mathbbmss{Q}} 53 | 54 | 55 | 56 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 57 | \newcommand*{\abs}[1]{| #1 |} 58 | 59 | 60 | 61 | \newtheorem{thm}{Theorem} 62 | \newtheorem{defn}{Definition} 63 | \newtheorem{prop}{Proposition} 64 | \newtheorem{lemma}{Lemma} 65 | \newtheorem{cor}{Corollary} 66 | \begin{document} 67 | The Latin phrase \emph{ad hoc} translates as ``toward this'', and is used in mathematics to describe anything that has been made up specifically for one particular purpose. 68 | 69 | \subsubsection*{Examples} 70 | \begin{enumerate} 71 | \item an ad hoc definition is a temporary definition that might 72 | be useful for the discussion at hand. For instance, let a \emph{nice set} 73 | be a closed set with smooth boundary. 74 | \item If a proof makes use of an ad hoc construction, the proof 75 | (typically) does not make use of a general theory for the problem. 76 | \end{enumerate} 77 | 78 | See also 79 | \PMlinkexternal{Article on ad hoc on Wikipedia}{http://en.wikipedia.org/wiki/Ad_hoc} 80 | %%%%% 81 | %%%%% 82 | \end{document} 83 | -------------------------------------------------------------------------------- /00-01-Expression.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Expression} 4 | \pmcreated{2013-03-22 17:50:11} 5 | \pmmodified{2013-03-22 17:50:11} 6 | \pmowner{Wkbj79}{1863} 7 | \pmmodifier{Wkbj79}{1863} 8 | \pmtitle{expression} 9 | \pmrecord{7}{40307} 10 | \pmprivacy{1} 11 | \pmauthor{Wkbj79}{1863} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | \pmrelated{Equation} 16 | \pmdefines{sum expression} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | \usepackage{pstricks} 24 | \usepackage{psfrag} 25 | \usepackage{graphicx} 26 | \usepackage{amsthm} 27 | %%\usepackage{xypic} 28 | 29 | \begin{document} 30 | An \emph{expression} is a symbol or \PMlinkescapetext{combination} of symbols used to denote a quantity or value. Expressions consist of constants, variables, operations, operators, functions, and parentheses.\\ 31 | 32 | \textbf{Note 1.} If an expression \PMlinkescapetext{contains} one or more operations to be performed in a certain \PMlinkname{order}{OrderOfOperations}, the expression may be named after the last (\PMlinkname{i.e.}{Ie} outermost) operation. For example, the expression $\displaystyle a^2-5\sqrt{a}+\frac{2}{3a}$ is a {\em sum expression}.\\ 33 | 34 | \textbf{Note 2.} An equation is a denoted equality of two expressions. 35 | 36 | \begin{thebibliography}{9} 37 | \bibitem{dict} \emph{Dictionary.com Unabridged} (version 1.1). Accessed on February 22, 2008. URL: \PMlinkexternal{http://dictionary.reference.com/browse/expression}{http://dictionary.reference.com/browse/expression} 38 | \end{thebibliography} 39 | %%%%% 40 | %%%%% 41 | \end{document} 42 | -------------------------------------------------------------------------------- /00-01-InfixNotation.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{InfixNotation} 4 | \pmcreated{2013-03-22 16:21:20} 5 | \pmmodified{2013-03-22 16:21:20} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{infix notation} 9 | \pmrecord{5}{38490} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | \pmsynonym{infix}{InfixNotation} 16 | \pmrelated{GeneralAssociativity} 17 | \pmdefines{infix arithmetic} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | 25 | %\usepackage{psfrag} 26 | %\usepackage{graphicx} 27 | %%%\usepackage{xypic} 28 | 29 | \begin{document} 30 | Infix notation is how we usually read and write arithmetic expressions. In this notation, the operator goes between the operands in the expression: 31 | 32 | $$ \text{(operand1)} \text{(operator)} \text{(operand2)} $$ 33 | 34 | E.g., $ 3 + 2 $, or $ 196 \times 11 $, etc. 35 | 36 | Infix notation suffers from some ambiguity; e.g. 37 | 38 | $$ 3 + 9 \times 2 $$ 39 | 40 | could mean $ ( 3 + 9) \times 2 $ or $ 3 + (9 \times 2) $. Parentheses are needed to specify the order of operations unambiguously. 41 | 42 | Postfix notation (or reverse-Polish notation) does not suffer this ambiguity; but it is considered harder for humans to read (hence its primary use in computer applications). 43 | 44 | The ``usual'' fix for the ambiguity problem described above is to provide a convention regarding precedence of operations. This is typically done for computer parsing of mathematical expressions rather than in math done by hand, because in the former case, the computer \emph{must} have some standard rules to proceed. For example, it is typical to make multiplication ``higher precedence'' than addition, so in the above case, $9 \times 2$ would be performed before adding the result to 3. 45 | 46 | The ambiguity problem only occurs when multiple operators are present in one expression, and thus, the associative law does not hold. E.g., there is no ambiguity in $1 + 2 + 3$, because $(1 + 2) + 3$ is the same as $1 + (2 + 3)$, by the associative property of addition. 47 | 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00-01-Rigid.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Rigid} 4 | \pmcreated{2013-03-22 14:38:10} 5 | \pmmodified{2013-03-22 14:38:10} 6 | \pmowner{matte}{1858} 7 | \pmmodifier{matte}{1858} 8 | \pmtitle{rigid} 9 | \pmrecord{11}{36219} 10 | \pmprivacy{1} 11 | \pmauthor{matte}{1858} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | \pmsynonym{rigidity result}{Rigid} 16 | \pmsynonym{rigidity theorem}{Rigid} 17 | \pmsynonym{rigidity}{Rigid} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | \usepackage{amsthm} 30 | 31 | \usepackage{mathrsfs} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | % 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | 46 | \newcommand{\sR}[0]{\mathbb{R}} 47 | \newcommand{\sC}[0]{\mathbb{C}} 48 | \newcommand{\sN}[0]{\mathbb{N}} 49 | \newcommand{\sZ}[0]{\mathbb{Z}} 50 | 51 | \usepackage{bbm} 52 | \newcommand{\Z}{\mathbbmss{Z}} 53 | \newcommand{\C}{\mathbbmss{C}} 54 | \newcommand{\R}{\mathbbmss{R}} 55 | \newcommand{\Q}{\mathbbmss{Q}} 56 | 57 | 58 | 59 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 60 | \newcommand*{\abs}[1]{| #1 |} 61 | 62 | 63 | 64 | \newtheorem{thm}{Theorem} 65 | \newtheorem{defn}{Definition} 66 | \newtheorem{prop}{Proposition} 67 | \newtheorem{lemma}{Lemma} 68 | \newtheorem{cor}{Corollary} 69 | \begin{document} 70 | Suppose $C$ is a collection of mathematical objects 71 | (for instance, sets, or functions). 72 | Then we say that $C$ is \emph{rigid} if every $c\in C$ 73 | is uniquely determined by less information about $c$ than 74 | one would expect. 75 | 76 | It should be emphasized that the above ``definition'' does not 77 | define a \emph{mathematical object}. Instead, it describes in what sense 78 | the adjective rigid is typically used in mathematics, 79 | by mathematicians. 80 | 81 | Let us illustrate this by some examples: 82 | 83 | \begin{enumerate} 84 | %\item Liouville's theorem: If 85 | %the difference of two entire functions is bounded, the functions are equal. 86 | \item Harmonic functions on the unit disk are rigid in the sense that 87 | they are uniquely determined by their boundary values. 88 | \item By the fundamental theorem of algebra, polynomials in $\sC$ 89 | are rigid in the sense that any polynomial is completely determined by 90 | its values on any countably infinite set, say $\sN$, or the unit disk. 91 | \item Linear maps $\mathscr{L}(X,Y)$ between vector spaces $X,Y$ 92 | are rigid in the sense that any $L\in\mathscr{L}(X,Y)$ is completely 93 | determined by its values on any set of basis vectors of $X$. 94 | \item Mostow's rigidity theorem 95 | \end{enumerate} 96 | %%%%% 97 | %%%%% 98 | \end{document} 99 | -------------------------------------------------------------------------------- /00-01-Strict.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Strict} 4 | \pmcreated{2013-03-22 14:45:23} 5 | \pmmodified{2013-03-22 14:45:23} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{strict} 9 | \pmrecord{12}{36397} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | \pmdefines{strict} 16 | \pmdefines{strictly} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | In mathematical writing, the adjective \emph{strict} is used in to modify technical \PMlinkescapetext{terms} which have \PMlinkescapetext{multiple} meanings. It indicates that the exclusive meaning of the term is to be understood. (More formally, one could say that this is the meaning which implies the other meanings.) 43 | 44 | This term is commonly used in the context of inequalities --- the phrases ``strictly less than'' and ``strictly greater than'' mean ``less than and not equal to'' and ``greater than and not equal to'', respectively. A related use occurs when comparing numbers to zero --- ``strictly positive'' and ``strictly negative'' mean ``positive and not equal to zero'' and ``negative and not equal to zero'', respectively. Also, in the context of functions, the adverb ``strictly '' is used to modify the terms ``monotonic'', ``increasing'', and ``decreasing''. 45 | 46 | On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons, one can use the phrases ``non-negative'', ``non-positive'', ``non-increasing'', and ``non-decreasing'' to make it clear that the inclusive sense of the terms is intended. 47 | 48 | Using such terminology helps avoid possible ambiguity and confusion. For instance, upon reading the phrase ``x is negative'', it is not immediately clear whether x = 0 is possible, since some authors consider zero to be positive while others consider zero not to be positive. Therefore, it is prudent to write either ``x is strictly negative'' or ``x is non-positive'' unless the distinction is unimportant or the context makes obvious which meaning is intended or it has been explicitly stated that the term "positive" is to be used in only one sense. (Here, in Planet Math, we have taken the third option by adopting the convention that zero is neither positive nor negative. Hence, the terms "strictly positive" and "positive" are synonyms here. However, the adverb "strictly" may still be necessary in some of the other contexts described avove.) 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /00-01-TextbookProjectsOnPlanetMath.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TextbookProjectsOnPlanetMath} 4 | \pmcreated{2013-03-22 14:46:20} 5 | \pmmodified{2013-03-22 14:46:20} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Textbook projects on PlanetMath} 9 | \pmrecord{7}{36417} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | %\pmkeywords{textbooks} 16 | 17 | \endmetadata 18 | 19 | 20 | \begin{document} 21 | This will be a guide to textbook projects on PM. Since I don't know 22 | if there currently are any textbook projects on PM, this is just a 23 | ``stub'' entry for now. If you create a textbook project, please add 24 | a link to it here. Let me know what other things you would like to 25 | see as part of this entry. (Eventually we will probably create a 26 | documentation item for textbook writing.) I think the 00-01 category 27 | can and should contain articles \emph{about} instructional exposition, 28 | just as well as examples of instructional exposition, so perhaps such 29 | discussions can begin here as well, and then branch off into other 30 | entries as that becomes convenient. 31 | 32 | An example of the sort of textbook we might want to have would 33 | be: a free book comparable to \emph{Introduction to Algorithms} by 34 | Cormen \emph{et al.}. 35 | 36 | \begin{itemize} 37 | \item Statistics on PlanetMath 38 | \end{itemize} 39 | 40 | %%%%% 41 | %%%%% 42 | \end{document} 43 | -------------------------------------------------------------------------------- /00-01-ToyTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ToyTheorem} 4 | \pmcreated{2013-03-22 13:55:35} 5 | \pmmodified{2013-03-22 13:55:35} 6 | \pmowner{matte}{1858} 7 | \pmmodifier{matte}{1858} 8 | \pmtitle{toy theorem} 9 | \pmrecord{7}{34684} 10 | \pmprivacy{1} 11 | \pmauthor{matte}{1858} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00-01} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \newcommand{\sR}[0]{\mathbb{R}} 41 | \newcommand{\sC}[0]{\mathbb{C}} 42 | \newcommand{\sN}[0]{\mathbb{N}} 43 | \newcommand{\sZ}[0]{\mathbb{Z}} 44 | 45 | % The below lines should work as the command 46 | % \renewcommand{\bibname}{References} 47 | % without creating havoc when rendering an entry in 48 | % the page-image mode. 49 | \makeatletter 50 | \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} 51 | \makeatother 52 | 53 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 54 | \newcommand*{\abs}[1]{| #1 |} 55 | \begin{document} 56 | A \emph{toy theorem} is a simplified version of a more general theorem. 57 | For instance, by introducing some simplifying assumptions in a theorem, 58 | one obtains a toy theorem. 59 | 60 | Usually, a toy theorem is used to illustrate the claim of a theorem. 61 | It can also be illustrative and insightful to study proofs of 62 | a toy theorem derived from a non-trivial theorem. 63 | Toy theorems also have a great education value. 64 | After presenting a theorem (with, say, a highly non-trivial proof), 65 | one can sometimes give some assurance that the theorem 66 | really holds, by proving a toy version of the theorem. 67 | 68 | For instance, a toy theorem of the Brouwer fixed point theorem 69 | is obtained by restricting the dimension to one. 70 | In this case, the Brouwer fixed point theorem follows 71 | almost immediately from the intermediate value theorem 72 | (see \PMlinkname{this page}{BrouwerFixedPointInOneDimension}). 73 | %%%%% 74 | %%%%% 75 | \end{document} 76 | -------------------------------------------------------------------------------- /00-02-PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PMPlanetaryUpdateOnDevelopmentalWebServersAndSitesincludingPlanetPhysicsorgWebServers} 4 | \pmcreated{2013-03-22 19:36:31} 5 | \pmmodified{2013-03-22 19:36:31} 6 | \pmowner{bci1}{20947} 7 | \pmmodifier{bci1}{20947} 8 | \pmtitle{PM Planetary Update on Developmental web Servers and Sites--including PlanetPhysics.org web servers} 9 | \pmrecord{31}{42602} 10 | \pmprivacy{1} 11 | \pmauthor{bci1}{20947} 12 | \pmcomment{trigger rebuild} 13 | \pmclassification{msc}{00-02} 14 | \pmclassification{msc}{00-01} 15 | 16 | \endmetadata 17 | 18 | \begin{document} 19 | \section{Web servers Updates:} 20 | (In reverse chronological order, with the first website/server listed being the latest one) 21 | Currently all servers are running at full speed... 22 | Books can be edited and saved in PDF format at the Media Wiki version 1.17 23 | \PMlinkexternal{PlanetPhysics.org website:}{http://wiki.planetphysics.info:8888} 24 | 25 | (Yes, you need to add :8888 after the URL for protection against spamming!) 26 | 27 | 28 | ``{\em Planetary.PlanetPhysics.info}'' is a new PM and PlanetPhysics.org developmental server/website for PlanetMath.org, running Drupal under Ubuntu 12.04.01 OS with {\em Planetary.PlanetMath.Beta3B}; it is currently dedicated as a Developmental PM new server for high storage backup and website development; it is also planned in the future to add to it a Planetary.PlanetPhysics website, running Drupal under Ubuntu 12.04.1 OS. This is currently the only PM developmental website that allows direct upload of Full (not filtered) HTML source code; 29 | 30 | {\bf Beta2} {\em "PlanetPhysics.org developmental website"} is actually an earlier PM developmental planetary website running on a different server 31 | 32 | {\bf Beta.PlanetMath.org} is running Beta3 Planetary for the PM developmental website with Drupal under Ubuntu 12.04.1 OS 33 | 34 | PlanetPhysics.org R- wiki is an Organizational Blog and Discussion R-wiki for backing up contents 35 | 36 | PlanetPhysics.org R-wiki on Noosphere 1.0 main server running under Ubuntu 10.04 37 | 38 | PlanetPhysics.org's MediaWiki v.1.17 server runs under Scientific Linux version 6.1 OS 39 | 40 | PlanetPhysics.us- is the Noosphere 1.0 website of PlanetPhysics.org 41 | 42 | PlanetPhysics.org 's \PMlinkexternal{Developmental-redux website}{http://pp-dev.org:8080/} runs on internet port 8080 Noosphere v. 1.5. 43 | 44 | 45 | \end{document} 46 | -------------------------------------------------------------------------------- /00A05-AnExampleOfMathematicalInduction.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AnExampleOfMathematicalInduction} 4 | \pmcreated{2013-03-22 15:39:46} 5 | \pmmodified{2013-03-22 15:39:46} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{an example of mathematical induction} 9 | \pmrecord{7}{37596} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{00A35} 16 | \pmclassification{msc}{03E25} 17 | \pmrelated{PrincipleOfFiniteInduction} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb,amscd} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | 25 | % used for TeXing text within eps files 26 | %\usepackage{psfrag} 27 | % need this for including graphics (\includegraphics) 28 | %\usepackage{graphicx} 29 | % for neatly defining theorems and propositions 30 | \usepackage{amsthm} 31 | % making logically defined graphics 32 | %%%\usepackage{xypic} 33 | 34 | % define commands here 35 | \begin{document} 36 | Below is a step-by-step demonstration of how mathematical induction (the Principle of Finite Induction) works. 37 | 38 | \textbf{Proposition.} For any positive integer $n$, $2^{n-1}\le n!$. 39 | 40 | \begin{proof} 41 | 42 | \textbf{Setup.} Let $S$ be the set of positive integers $n$ satisfying the rule: $2^{n-1}\le n!$. We want to show that $S$ is the set of \emph{all} positive integers, which would prove our proposition. 43 | 44 | \textbf{Initial Step.} For $n=1$, $2^{n-1}=2^{1-1}=2^0=1$, while $n!=1!=1$. So $2^{n-1}=n!$ for $n=1$ and thus $2^{n-1}\le n!$ all the more so. This shows that $1\in S$. 45 | 46 | \textbf{Induction Step 1.} Assume that for $n=k$, $k$ a positive integer, $2^{n-1}\le n!$. In other words, we assume that $k\in S$, or that $2^{k-1}\le k!$. 47 | 48 | \textbf{Induction Step 2.} Next, we want to show that $k+1\in S$. If we let $n=k+1$, then by the assumption of the proposition, showing $n=k+1\in S$ is the same as showing $2^{n-1}\le n!$, or $2^k\le (k+1)!$. This can be done by the following calculation: 49 | \begin{eqnarray} 50 | 2^k &=& 2^{k-1}2\\ &\le& k!2\\ &\le& k!(k+1)\\ &=& (k+1)!, 51 | \end{eqnarray} 52 | where Equations (1) and (4) are just definitions of the power and the factorial of a number, respectively. Step (3) is the fact that $2\le k+1$ for any positive integer $k$ (which, incidentally, can be proved by mathematical induction as well). Step (2) follows from the \emph{induction step}, the assumption that we made in the \textbf{Induction Step 1.} in the previous paragraph. Because $2^k\le (k+1)!$, we have thus shown that $n=k+1\in S$, proving the proposition. 53 | \end{proof} 54 | 55 | \textbf{Remark.} All these steps are essential in any proof by (mathematical) induction. However, in more advanced math articles and books, some or all of these steps are omitted with the assumption that the reader is already familiar with the concepts and the steps involved. 56 | %%%%% 57 | %%%%% 58 | \end{document} 59 | -------------------------------------------------------------------------------- /00A05-ArgMinAndArgMax.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ArgMinAndArgMax} 4 | \pmcreated{2013-03-22 14:27:55} 5 | \pmmodified{2013-03-22 14:27:55} 6 | \pmowner{kshum}{5987} 7 | \pmmodifier{kshum}{5987} 8 | \pmtitle{arg min and arg max} 9 | \pmrecord{11}{35986} 10 | \pmprivacy{1} 11 | \pmauthor{kshum}{5987} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmdefines{argmin argmax} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | %\usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | For a real-valued function $f$ with domain $S$, $\arg \min_{x \in S} f(x)$ is the set of elements in $S$ that achieve the global minimum in $S$, 42 | \[ 43 | {\arg \min}_{x \in S} f(x) = \{ x \in S :\, f(x) = \min_{y\in S} f(y) \}. 44 | \] 45 | 46 | $\arg \max_{x \in S} f(x)$ is the set of elements in $S$ that achieve the global maximum in $S$, 47 | \[ 48 | {\arg \max}_{x \in S} f(x) = \{ x \in S :\, f(x) = \max_{y\in S} f(y) \}. 49 | \] 50 | 51 | 52 | 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /00A05-ArithmeticProgression.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ArithmeticProgression} 4 | \pmcreated{2013-03-22 13:39:00} 5 | \pmmodified{2013-03-22 13:39:00} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{arithmetic progression} 9 | \pmrecord{10}{34302} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{11B25} 16 | \pmrelated{MulidimensionalArithmeticProgression} 17 | \pmrelated{SumOfKthPowersOfTheFirstNPositiveIntegers} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | \begin{document} 25 | \PMlinkescapeword{columns} 26 | Arithmetic progression of length $n$, initial term $a_1$ and common difference $d$ is the sequence $a_1, a_1+d,a_1+2d,\dotsc,a_1+(n-1)d$. 27 | 28 | %%%This code is not displayed by PlanetMath... seems to be a bug in LaTeX2HTML 29 | %%% Added later: parbox fixes it since it forces LaTeX2HTML to render the whole 30 | %%% thing as an image 31 | The sum of terms of an arithmetic progression can be computed using Gauss's trick: 32 | 33 | \parbox{\linewidth}{\begin{align*} 34 | S&=\makebox[7em]{$(a_1+0)$}+\makebox[7em]{$(a_1+d)$}+\dotsb+\makebox[7em]{$(a_1+(n-2)d)$} 35 | +\makebox[7em]{$(a_1+(n-1)d)$}\\ 36 | +\underline{S\vphantom{\makebox[7em]{$(a_1+(n-1)d)$}}}&\underline{{}=\makebox[7em]{$(a_1+(n-1)d)$}+ 37 | \makebox[7em]{$(a_1+(n-2)d)$}+\dotsb+\makebox[7em]{$(a_1+d)$} 38 | +\makebox[7em]{$(a_1+0)$}}\\ 39 | 2S&=\makebox[7em]{$(2a_1+(n-1)d)$}+\makebox[7em]{$(2a_1+(n-1)d)$}+\dotsb+\makebox[7em]{$(2a_1+(n-1)d)$}+ 40 | \makebox[7em]{$(2a_1+(n-1)d)$}. 41 | \end{align*}} 42 | 43 | We just add the sum with itself written backwards, and the sum of each of the columns equals to $(2a_1+(n-1)d)$. The sum is then 44 | \begin{equation*} 45 | S=\frac{(2a_1+(n-1)d)n}{2}. 46 | \end{equation*} 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /00A05-ArithmeticgeometricharmonicMeansInequality.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ArithmeticgeometricharmonicMeansInequality} 4 | \pmcreated{2013-03-22 11:42:32} 5 | \pmmodified{2013-03-22 11:42:32} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{arithmetic-geometric-harmonic means inequality} 9 | \pmrecord{22}{30025} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{20-XX} 16 | \pmclassification{msc}{26D15} 17 | \pmsynonym{harmonic-geometric-arithmetic means inequality}{ArithmeticgeometricharmonicMeansInequality} 18 | \pmsynonym{arithmetic-geometric means inequality}{ArithmeticgeometricharmonicMeansInequality} 19 | \pmsynonym{AGM inequality}{ArithmeticgeometricharmonicMeansInequality} 20 | \pmsynonym{AGMH inequality}{ArithmeticgeometricharmonicMeansInequality} 21 | %\pmkeywords{inequality} 22 | %\pmkeywords{mean} 23 | %\pmkeywords{arithmetic mean} 24 | %\pmkeywords{geometric mean} 25 | %\pmkeywords{harmonic mean} 26 | \pmrelated{ArithmeticMean} 27 | \pmrelated{GeometricMean} 28 | \pmrelated{HarmonicMean} 29 | \pmrelated{GeneralMeansInequality} 30 | \pmrelated{WeightedPowerMean} 31 | \pmrelated{PowerMean} 32 | \pmrelated{RootMeanSquare3} 33 | \pmrelated{ProofOfGeneralMeansInequality} 34 | \pmrelated{JensensInequality} 35 | \pmrelated{DerivationOfHarmonicMeanAsTheLimitOfThePowerMean} 36 | \pmrelated{MinimalAndMaximalNumber} 37 | \pmrelated{ProofOfArithm} 38 | 39 | \endmetadata 40 | 41 | \usepackage{amssymb} 42 | \usepackage{amsmath} 43 | \usepackage{amsfonts} 44 | \usepackage{graphicx} 45 | %%%%%%%%%%%%%%%%\usepackage{xypic} 46 | \begin{document} 47 | Let $x_1,x_2,\ldots,x_n$ be positive numbers. 48 | Then 49 | \begin{eqnarray*} 50 | \max\{x_1,x_2,\ldots,x_n\} &\ge& \frac{x_1+x_2+\cdots+x_n}{n}\\ 51 | &\ge& \sqrt[n]{x_1 x_2\cdots x_n} \\ 52 | &\ge& \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\\ 53 | &\ge& \min\{x_1,x_2,\ldots,x_n\} 54 | \end{eqnarray*} 55 | 56 | The equality is obtained if and only if $x_1=x_2=\cdots = x_n$. 57 | 58 | There are several generalizations to this inequality using power means and weighted power means. 59 | %%%%% 60 | %%%%% 61 | %%%%% 62 | %%%%% 63 | %%%%% 64 | %%%%% 65 | %%%%% 66 | %%%%% 67 | %%%%% 68 | %%%%% 69 | %%%%% 70 | %%%%% 71 | %%%%% 72 | %%%%% 73 | %%%%% 74 | %%%%% 75 | \end{document} 76 | -------------------------------------------------------------------------------- /00A05-Arity.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Arity} 4 | \pmcreated{2013-03-22 13:00:01} 5 | \pmmodified{2013-03-22 13:00:01} 6 | \pmowner{Henry}{455} 7 | \pmmodifier{Henry}{455} 8 | \pmtitle{arity} 9 | \pmrecord{4}{33378} 10 | \pmprivacy{1} 11 | \pmauthor{Henry}{455} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{ary}{Arity} 16 | \pmsynonym{-arity}{Arity} 17 | \pmrelated{BinaryOperation} 18 | \pmdefines{unary} 19 | \pmdefines{binary} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | %\PMlinkescapeword{theory} 45 | \begin{document} 46 | The \emph{arity} of something is the number of arguments it takes. This is usually applied to functions: an $n$-ary function is one that takes $n$ arguments. \emph{Unary} is a synonym for $1$-ary, and binary for $2$-ary. 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /00A05-Calculator.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Calculator} 4 | \pmcreated{2013-03-22 16:39:16} 5 | \pmmodified{2013-03-22 16:39:16} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{calculator} 9 | \pmrecord{4}{38859} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A07} 16 | \pmrelated{CalculatorAndCASSupportForVariousPositionalBases} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | 42 | \begin{document} 43 | A {\em calculator} is an electronic, electrical or mechanical device (hardware) or a computer program (software) designed to perform a predefined set of arithmetic computations on numbers entered by the user and display the results. 44 | 45 | The abacus is sometimes given as an example of an early calculator; however, the completion of a calculation requires the active participation of the user and is easily subject to user error even when the device passes a diagnostic with flying colors, whereas a modern calculator in working order calculates and displays the correct result regardless of whether or not the user sticks around to see it (this is subject to certain caveats regarding precision, however). Of course it has always been the case in the history of calculators that users may enter incorrect inputs or misunderstand the operation of the device and thus get irrelevant results. 46 | 47 | One of the first mechanical calculators was designed by Blaise Pascal in the 17th Century. It used a gear for each digit and the gears were connected, it took up a little space on a desk. It was limited to addition and subtraction. By the 1990s, electronic calculators were ubiquitous and small enough to put in a pocket. 48 | 49 | A basic calculator has about twenty keys: the digits 0 to 9, a decimal point, sign change, addition, subtraction, multiplication, division, equal or Enter key and a C key to clear error exceptions. Occasionally such calculators also have keys for percentage and square root. Another option sometimes found on basic calculators is a memory register and associated keys M+, MR and MC (for adding to memory register, recalling memory and clearing memory, respectively). 50 | 51 | More advanced calculators include scientific calculators, graphing calculators, programmable calculators. Most calculators use standard infix notation, but there are reverse Polish notation calculators also available on the market. 52 | %%%%% 53 | %%%%% 54 | \end{document} 55 | -------------------------------------------------------------------------------- /00A05-Characterisation.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Characterisation} 4 | \pmcreated{2013-03-22 14:22:28} 5 | \pmmodified{2013-03-22 14:22:28} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{characterisation} 9 | \pmrecord{18}{35865} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{characterization}{Characterisation} 16 | \pmsynonym{defining property}{Characterisation} 17 | \pmrelated{AlternativeDefinitionOfGroup} 18 | \pmrelated{EquivalentFormulationsForContinuity} 19 | \pmrelated{MultiplicationRuleGivesInverseIdeal} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | \begin{document} 45 | In mathematics, {\em characterisation} usually means a property or a condition to define a certain notion. \,A notion may, under some presumptions, have different \PMlinkescapetext{equivalent} ways to define it. 46 | 47 | For example, let $R$ be a commutative ring with non-zero unity (the presumption). \,Then the following are equivalent: 48 | 49 | (1) All finitely generated regular ideals of $R$ are invertible. 50 | 51 | (2) The \PMlinkescapetext{formula} \,$(a,\,b)(c,\,d) = (ac,\,bd,\,(a+b)(c+d))$\, for multiplying ideals of $R$ is valid always when at least one of the elements $a$, $b$, $c$, $d$ of $R$ is not zero-divisor. 52 | 53 | (3) Every overring of $R$ is integrally closed. 54 | 55 | Each of these conditions is sufficient (and necessary) for characterising and defining the Pr\"ufer ring. 56 | %%%%% 57 | %%%%% 58 | \end{document} 59 | -------------------------------------------------------------------------------- /00A05-DecimalPoint.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{DecimalPoint} 4 | \pmcreated{2013-03-22 17:23:33} 5 | \pmmodified{2013-03-22 17:23:33} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{decimal point} 9 | \pmrecord{7}{39760} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | A {\em decimal point} is a symbol separating those digits representing integer powers of a base (usually base 10) on the left, and those representing fractional powers of a base (the base raised to a negative number) on the right. For example, in $\pi \approx 3.14$, the 3 to the left of the decimal point corresponds to $3 \times 10^0$, while the 1 to the right of the decimal point corresponds to $1 \times 10^{-1}$. 42 | 43 | Most scientific calculators capable of displaying binary, octal and hexadecimal limit numbers in those bases to integers, making moot the issue of what to call the decimal point in those bases. 44 | 45 | The decimal point is generally omitted for integers. However, Mathematica will use the decimal point at the end of an integer to indicate the value has been computed using floating-point arithmetic and loss of precision is possible. For example, \verb=(1/2)^(-1)= gives ``2'' as an answer but \verb=.5^(-1)= gives ``2.'' for the answer. Even more pointedly, \verb=1 + 1= gives ``2'' as the answer but \verb=1. + 1.= gives ``2.'' as the answer. 46 | 47 | In the United States, the decimal point is usually aligned with the bottom of the digit glyphs, while in the United Kingdom it is usually centered (and is distinguished from the central dot multiplication operator purely on spacing). In Europe, a comma is used instead, so our example would be written $\pi \approx 3,\!14$. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A05-FundamentalTheoremsInMathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{FundamentalTheoremsInMathematics} 4 | \pmcreated{2013-03-22 19:11:13} 5 | \pmmodified{2013-03-22 19:11:13} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{fundamental theorems in mathematics} 9 | \pmrecord{11}{42095} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Topic} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | %\pmkeywords{fundamental theorem} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | \usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \theoremstyle{definition} 42 | \newtheorem*{thmplain}{Theorem} 43 | 44 | \begin{document} 45 | In different areas of mathematics, there are certain \PMlinkescapetext{central} theorems called \emph{fundamental}.\, We list a number of them below. 46 | 47 | \begin{itemize} 48 | \item fundamental cohomology theorem 49 | \item fundamental homomorphism theorem 50 | \item fundamental theorem of algebra 51 | \item fundamental theorem of arithmetics 52 | \item fundamental theorem of ideal theory 53 | \item fundamental theorem of calculus 54 | \item fundamental theorem of calculus for Kurzweil-Henstock integral 55 | \item fundamental theorem of calculus for Lebesgue integration 56 | \item fundamental theorem of calculus for Riemann integration 57 | \item \PMlinkid{fundamental theorem of calculus of variations}{6745} 58 | \item fundamental theorem of complex analysis 59 | \item fundamental theorem of demography 60 | \item fundamental theorem of finitely generated Abelian groups 61 | \item fundamental theorem of Galois theory 62 | \item fundamental theorem of Gaussian quadrature 63 | \item fundamental theorem of integral calculus 64 | \item fundamental theorem on isogonal lines 65 | \item fundamental theorem of linear algebra 66 | \item fundamental theorem of linear programming 67 | \item \PMlinkid{fundamental theorem of line integrals}{12096} 68 | \item fundamental theorem of plane curves 69 | \item fundamental theorem of projective geometry 70 | \item fundamental theorem of Riemannian geometry 71 | \item fundamental theorem of space curves 72 | \item fundamental theorem of symmetric polynomials 73 | \item fundamental theorem of topology 74 | \item fundamental theorem of topos theory 75 | \item fundamental theorem of transcendence 76 | \item \PMlinkid{fundamental theorem of vector calculus}{10506} 77 | 78 | \end{itemize} 79 | %%%%% 80 | %%%%% 81 | \end{document} 82 | -------------------------------------------------------------------------------- /00A05-Grapher.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Grapher} 4 | \pmcreated{2013-03-22 17:46:13} 5 | \pmmodified{2013-03-22 17:46:13} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Grapher} 9 | \pmrecord{4}{40226} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A07} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | 31 | % need this for including graphics (\includegraphics) 32 | \usepackage{graphicx} 33 | 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \begin{document} 44 | {\em Grapher} is a software graphing calculator that comes bundled with the Apple Mac OS Xoperating system. It can graph 2-dimensional equations like $y = \sin x$, as well as 3-dimensional equations like $z = \frac{y^3}{x^2 + y^2}$ (Cartan's umbrella, illustrated below). 45 | 46 | \begin{center} 47 | \includegraphics{C:TempGrapherScreenShot} 48 | \end{center} 49 | 50 | Equations to be graphed can be entered in Cartesian form or in parametric form, polar coordinates, logarithmic, etc. The program automatically ``typesets'' the formulas as they are entered, for example, converting \verb=x^2= to $x^2$ or \verb=1/x= to $\frac{1}{x}$. The program can create animations, and one can also move a graph or rotate it (just by dragging it, for 3D graphs; for 2D graphs one must first select the Move Tool). 51 | 52 | In a default installation, the program is in the Utilities subfolder of the Applications folder. Mac OS 9 had a similar program, called Graphing Calculator, which was also capable of graphing 2D and 3D equations as well as animations. 53 | 54 | \begin{thebibliography}{2} 55 | \bibitem{ml} Maria Langer, {\it Visual Quickstart Guide: Mac OS 9.1}. New York: Peachpit Press (2001): 107 56 | \bibitem{dp} David Pogue \& Adam Goldstein, {\it The Missing Manual: Switching to the Mac, Tiger Edition}. Sebastopol: O'Reilly Media, Inc. (2005): 452 - 453 57 | \end{thebibliography} 58 | %%%%% 59 | %%%%% 60 | \end{document} 61 | -------------------------------------------------------------------------------- /00A05-Hypothesis.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Hypothesis} 4 | \pmcreated{2013-03-22 17:15:18} 5 | \pmmodified{2013-03-22 17:15:18} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{hypothesis} 9 | \pmrecord{6}{39590} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | \PMlinkescapeword{support} 42 | \PMlinkescapeword{field} 43 | \PMlinkescapeword{label} 44 | 45 | In mathematics, a {\em hypothesis} is an unproven statement which is supported by all the available data and by many weaker results. An unproven mathematical statement is usually called a ``conjecture'', and while experimentation can sometimes produce millions of examples to support a conjecture, usually nothing short of a proof can convince experts in the field. But when a conjecture is supported not only but all the available data but also by numerous weaker results, it is upgraded in label to a hypothesis. The most famous conjecture in mathematics is the Riemann hypothesis, which despite many attempts at a proof, is supported by many related results. The convexity conjecture, on the other hand, is considered ``incompatible'' with the $n$-tuples conjecture and more results appear to support the latter, thus neither is upgraded to hypothesis. 46 | 47 | \begin{thebibliography}{1} 48 | \bibitem{rc} R. Crandall \& C. Pomerance, {\it Prime Numbers: A Computational Perspective}, Springer, NY, 2001: 1.2.4 49 | \end{thebibliography} 50 | %%%%% 51 | %%%%% 52 | \end{document} 53 | -------------------------------------------------------------------------------- /00A05-Indeterminate.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Indeterminate} 4 | \pmcreated{2013-03-22 14:47:33} 5 | \pmmodified{2013-03-22 14:47:33} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{indeterminate} 9 | \pmrecord{5}{36444} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmrelated{Parameter} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb,amscd} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | \PMlinkescapeword{solvable} 42 | \PMlinkescapeword{alphabet} 43 | 44 | An {\em indeterminate}\, is simply a variable that is not known or solvable. \,It is usually denoted by a mathematical alphabet ($x$, $y$, $z$, or $\alpha$, $\beta$, etc...). \,It is important to distinguish between a variable and an indeterminate in that a variable is solvable, at least conditionally. \,To make this more precise, let's see two examples: 45 | 46 | \begin{enumerate} 47 | \item Let $x$ be a variable such that \,$2+3x=a+bx$, where $a,b\in\mathbb{Q}$. Then \,$x=(a-2)/(3-b)$. \,Here $x$ is solvable conditioned on the equation given. Any values of $a$ and $b\,(\neq 3)$ will yield a value for $x$. 48 | \item Let $x$ be an indeterminate such that \,$2+3x=a+bx$, where \,$a,\,b\in\mathbb{Q}$. \,Since $x$ can not be solved, we have \,$2=a$\, and \,$3=b$. \,Note that if $a$ and $b$ are previously assigned to be values other than 2 and 3 respectively, then $x$ is no longer an indeterminate. 49 | \end{enumerate} 50 | %%%%% 51 | %%%%% 52 | \end{document} 53 | -------------------------------------------------------------------------------- /00A05-IntegralSign.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{IntegralSign} 4 | \pmcreated{2013-03-22 18:04:00} 5 | \pmmodified{2013-03-22 18:04:00} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{integral sign} 9 | \pmrecord{7}{40599} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{00A06} 16 | \pmrelated{RiemannIntegral} 17 | \pmrelated{RiemannStieltjesIntegral} 18 | \pmrelated{Integral2} 19 | \pmdefines{integrand} 20 | \pmdefines{integrate} 21 | 22 | \endmetadata 23 | 24 | % this is the default PlanetMath preamble. as your knowledge 25 | % of TeX increases, you will probably want to edit this, but 26 | % it should be fine as is for beginners. 27 | 28 | % almost certainly you want these 29 | \usepackage{amssymb} 30 | \usepackage{amsmath} 31 | \usepackage{amsfonts} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | \usepackage{amsthm} 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | 46 | \theoremstyle{definition} 47 | \newtheorem*{thmplain}{Theorem} 48 | 49 | \begin{document} 50 | The {\em integral sign} 51 | $$\int$$ 52 | is a stylised version of the {\em long s} letter. 53 | 54 | The long s is a typographic variant of lowercase s, being the only lowercase s in the Carolingian minuscule script.\, The 55 | modern short (round) s appeared later to the ends of words, and has now replaced completely the long s in the antiqua script. 56 | 57 | Gottfried Wilhelm Leibniz introduced the integral sign as the first letter s of the Latin word {\em summa} (`sum').\, The long shape of $\displaystyle\int$ may be thought to symbolically depict the fact that \PMlinkname{integral}{DefiniteIntegral} is a limiting case of sum.\\ 58 | 59 | A variant 60 | $$\oint$$ 61 | of the integral sign is used in integrals taken along a closed curve in $\mathbb{R}^2$ or about a closed surface in $\mathbb{R}^3$; see e.g. Cauchy integral theorem, derivation of heat equation.\\ 62 | 63 | The function given after the integral sign, i.e. the function to be {\em integrated}, is the {\em integrand}. 64 | %%%%% 65 | %%%%% 66 | \end{document} 67 | -------------------------------------------------------------------------------- /00A05-Introducing0thPower.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Introducing0thPower} 4 | \pmcreated{2013-03-22 13:24:20} 5 | \pmmodified{2013-03-22 13:24:20} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{introducing 0th power} 9 | \pmrecord{8}{33948} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Topic} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmrelated{EmptyProduct} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | Let $a$ be a number not equal to zero. Then for all $n \in \mathbb{N}$, we have that $a^n$ is the product of $a$ with itself $n$ \PMlinkescapetext{times}. Using the fact that the integer 1 is a multiplicative identity, ($a\cdot 1=a$ for any $a$), we can write 42 | \begin{displaymath} 43 | a^n \cdot 1=a^n=a^{n+0}=a^n\cdot a^0, 44 | \end{displaymath} 45 | where we have used the properties of exponents under multiplication. Now, after canceling a factor of $a^n$ from both sides of the above equation, we derive that $a^0=1$ for any non-zero number. 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /00A05-KnuthsUpArrowNotation.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{KnuthsUpArrowNotation} 4 | \pmcreated{2013-03-22 12:58:43} 5 | \pmmodified{2013-03-22 12:58:43} 6 | \pmowner{Henry}{455} 7 | \pmmodifier{Henry}{455} 8 | \pmtitle{Knuth's up arrow notation} 9 | \pmrecord{7}{33350} 10 | \pmprivacy{1} 11 | \pmauthor{Henry}{455} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{up-arrow}{KnuthsUpArrowNotation} 16 | \pmsynonym{up arrow}{KnuthsUpArrowNotation} 17 | \pmsynonym{up-arrow notation}{KnuthsUpArrowNotation} 18 | \pmsynonym{up arrow notation}{KnuthsUpArrowNotation} 19 | \pmsynonym{Knuth notation}{KnuthsUpArrowNotation} 20 | \pmrelated{ConwaysChainedArrowNotation} 21 | 22 | \endmetadata 23 | 24 | % this is the default PlanetMath preamble. as your knowledge 25 | % of TeX increases, you will probably want to edit this, but 26 | % it should be fine as is for beginners. 27 | 28 | % almost certainly you want these 29 | \usepackage{amssymb} 30 | \usepackage{amsmath} 31 | \usepackage{amsfonts} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | %\usepackage{amsthm} 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | %\PMlinkescapeword{theory} 46 | \begin{document} 47 | \emph{Knuth's up arrow noation} is a way of writing numbers which would be unwieldy in standard decimal notation. It expands on the exponential notation $m\uparrow n=m^n$. Define $m\uparrow\uparrow 0=1$ and $m \uparrow\uparrow n=m\uparrow(m\uparrow\uparrow [n-1])$. 48 | 49 | Obviously $m\uparrow\uparrow 1=m^1=m$, so $3\uparrow\uparrow 2=3^{3\uparrow\uparrow 1}=3^3=27$, but $2\uparrow\uparrow 3=2^{2 \uparrow\uparrow 2}=2^{2^{2\uparrow\uparrow 1}}=2^{(2^2)}=16$. 50 | 51 | In general, $m\uparrow\uparrow n=m^{m^{\cdots^m}}$, a tower of height $n$. 52 | 53 | Clearly, this process can be extended: $m\uparrow\uparrow\uparrow 0=1$ and $m\uparrow\uparrow\uparrow n=m\uparrow\uparrow(m\uparrow\uparrow\uparrow [n-1])$. 54 | 55 | An alternate notation is to write $m^{(i)}n$ for $m\underbrace{\uparrow\cdots\uparrow}_{i-2 \text{~times}}n$. ($i-2$ times because then $m^{(2)}n=m\cdot n$ and $m^{(1)}n=m+n$.) Then in general we can define $m^{(i)}n=m^{(i-1)}(m^{(i)}(n-1))$. 56 | 57 | To get a sense of how quickly these numbers grow, $3\uparrow\uparrow\uparrow 2=3\uparrow\uparrow 3$ is more than seven and a half trillion, and the numbers continue to grow much more than exponentially. 58 | %%%%% 59 | %%%%% 60 | \end{document} 61 | -------------------------------------------------------------------------------- /00A05-MacOSCalculator.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MacOSCalculator} 4 | \pmcreated{2013-03-22 16:39:26} 5 | \pmmodified{2013-03-22 16:39:26} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Mac OS Calculator} 9 | \pmrecord{6}{38862} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A07} 16 | \pmdefines{Mac OS 9 Calculator} 17 | \pmdefines{Mac OS X Calculator} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \begin{document} 44 | The {\em Mac OS Calculator} is a software calculator that comes bundled with the Apple Mac OS operating system. As late as Mac OS 9, the Calculator was a very basic calculator with only the arithmetic operations. In Mac OS X, the Calculator program was upgraded, with not just the addition of Scientific mode (a scientific calculator) but also Programmer mode with functions useful for computer programmers, such as Unicode character lookup by numerical value. The currently displayed value is not lost at a change of mode. 45 | 46 | Like the Windows Calculator, for the Mac OS Calculator $0^0 = 1$. 47 | 48 | \begin{thebibliography}{1} 49 | \bibitem{ml} Maria Langer, {\it Mac OS 8: Visual Quickstart Guide} Berkeley: Peachpit Press (1997): 108 50 | \end{thebibliography} 51 | %%%%% 52 | %%%%% 53 | \end{document} 54 | -------------------------------------------------------------------------------- /00A05-MockupOfABasicCalculator.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MockupOfABasicCalculator} 4 | \pmcreated{2013-03-22 16:53:48} 5 | \pmmodified{2013-03-22 16:53:48} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{mock-up of a basic calculator} 9 | \pmrecord{4}{39153} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A07} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | This mock-up of a basic calculator is realistic in that it has almost every function one can expect on a typical basic calculator. The layout will of course be different on an actual calculator. A calculator shaped like, say, a credit card will have more columns of buttons than rows. 43 | 44 | \begin{tabular}{|c|c|c|c|c|} 45 | MC & & OFF & C & ON \\ 46 | MR & & & & $\div$ \\ 47 | M+ & 7 & 8 & 9 & $\times$ \\ 48 | STO & 4 & 5 & 6 & $-$ \\ 49 | $\sqrt{x}$ & 1 & 2 & 3 & + \\ 50 | \% & 0 & . & $\pm$ & = \\ 51 | \end{tabular} 52 | 53 | A basic calculator can be counted on to have buttons for the basic arithmetic operations (addition, subtraction, multiplication and division). A button for square root is sometimes provided, its operation usually being ``postfix''. The percentage key, when provided, does not always behave in a standard way. For example, on some calculators, a 15\% tip on a meal costing $\$42.54$ might be computed as $[4] [2] [.] [5] [4] [+] [1] [5] [\%] [=]$. 54 | %%%%% 55 | %%%%% 56 | \end{document} 57 | -------------------------------------------------------------------------------- /00A05-MockupOfScientificCalculator.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MockupOfScientificCalculator} 4 | \pmcreated{2013-03-22 16:53:50} 5 | \pmmodified{2013-03-22 16:53:50} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{mock-up of scientific calculator} 9 | \pmrecord{4}{39154} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A65} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | This mock-up of a scientific calculator is realistic in that it has almost every function one can expect on a typical scientific calculator. It is unrealistic in that each function gets its own key. Usually, scientific calculators have some kind of shift key (labeled ``Shift'', ``2nd'' or something similar) and almost all the other buttons (including the digit buttons) have a second or even third use. Sometimes these shifts make sense (sine and arcsine on the same key, for example), sometimes less so (for example, the random number generator on the key for $\pi$ or the decimal point). 43 | 44 | \begin{tabular}{|c|c|c|c|c|} 45 | BIN & EE & & OFF & ON \\ 46 | OCT & $\frac{d}{c}$ & RAND & & C \\ 47 | DEC & $a \frac{b}{c}$ & $\Sigma +$ & $\Sigma -$ & $y\sigma n - 1$ \\ 48 | HEX & nCr & $\hat{x}$ & $\hat{y}$ & $y\sigma n$ \\ 49 | x! & nPr & $\bar{x}$ & $\bar{y}$ & $x\sigma n - 1$ \\ 50 | $e^x$ & $\pi$ & $P \to R$ & $R \to P$ & $x\sigma n$ \\ 51 | ln & DEG & GRAD & RAD & MR \\ 52 | $10^x$ & $\sin^{-1}$ & $\cos^{-1}$ & $\tan^{-1}$ & M+ \\ 53 | log & sin & cos & tan & STO \\ 54 | $\sqrt[y]{x}$ & D & E & F & \% \\ 55 | $x^y$ & A & B & C & $\div$ \\ 56 | $\sqrt[3]{x}$ & 7 & 8 & 9 & $\times$ \\ 57 | $x^3$ & 4 & 5 & 6 & $-$ \\ 58 | $\sqrt{x}$ & 1 & 2 & 3 & + \\ 59 | $x^2$ & 0 & . & $\pm$ & = \\ 60 | $\frac{1}{x}$ & & & & \\ 61 | \end{tabular} 62 | 63 | Not all scientific calculators support binary, octal and hexadecimal display. Fractions display and conversion is another category of functions that is available on many, but not all, scientific calculators. The trigonometric and statistical functions, on the other hand, are always standard, even if the button labels aren't always (mainly for the statistical functions). The percentage key is more of a rarity on scientific calculators, something reflected by the Windows Calculator, which has percentage in Standard mode but not Scientific (puzzlingly, this is also true of the square root key). 64 | 65 | Square root and cubic root are usually ``postfix'' operations, e.g., meaning that to compute $\sqrt{2209}$ one would enter $[2] [2] [0] [9] [\sqrt{x}]$. On the CVS-brand scientific calculator with 2-line display, however, that would result in a ``syntax error''; the square root key has to be pushed before the digits of the operand. 66 | %%%%% 67 | %%%%% 68 | \end{document} 69 | -------------------------------------------------------------------------------- /00A05-Parametre.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Parametre} 4 | \pmcreated{2013-03-22 17:06:59} 5 | \pmmodified{2013-03-22 17:06:59} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{parametre} 9 | \pmrecord{17}{39417} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{parameter}{Parametre} 16 | \pmrelated{Indeterminate} 17 | \pmrelated{DerivativeForParametricForm} 18 | \pmrelated{Curve} 19 | \pmrelated{PerimeterOfAstroid} 20 | \pmrelated{CissoidOfDiocles} 21 | \pmrelated{Variable} 22 | \pmrelated{SurfaceNormal} 23 | \pmdefines{auxiliary variable} 24 | \pmdefines{parametric form} 25 | \pmdefines{parametric presentation} 26 | \pmdefines{parameter of parabola} 27 | 28 | \endmetadata 29 | 30 | % this is the default PlanetMath preamble. as your knowledge 31 | % of TeX increases, you will probably want to edit this, but 32 | % it should be fine as is for beginners. 33 | 34 | % almost certainly you want these 35 | \usepackage{amssymb} 36 | \usepackage{amsmath} 37 | \usepackage{amsfonts} 38 | 39 | % used for TeXing text within eps files 40 | %\usepackage{psfrag} 41 | % need this for including graphics (\includegraphics) 42 | %\usepackage{graphicx} 43 | % for neatly defining theorems and propositions 44 | \usepackage{amsthm} 45 | % making logically defined graphics 46 | %%%\usepackage{xypic} 47 | 48 | % there are many more packages, add them here as you need them 49 | 50 | % define commands here 51 | 52 | \theoremstyle{definition} 53 | \newtheorem*{thmplain}{Theorem} 54 | 55 | \begin{document} 56 | \PMlinkescapeword{constant} 57 | \emph{Parametre} means often a quantity which is considered as constant in a certain situation but which may take different values in other situations; so the parametre is a ``variable constant''.\; But in giving a curve or a surface in {\em parametric form}, the parametres work as proper variables which determine the values of the coordinates of the points; then we can describe the parametres as ``auxiliary variables''. 58 | 59 | The parametric \PMlinkescapetext{presentation} 60 | \begin{align*} 61 | \begin{cases} 62 | x = a\cos{t}\\ 63 | y = a\sin{t} 64 | \end{cases} 65 | \end{align*} 66 | of the origin-centered circle \PMlinkescapetext{contains both above-mentioned sorts} of parametres:\; $a$ (the radius) is a variable constant which is held constant all the time when one considers one circle;\, $t$ is an auxiliary variable which has to get all real values (e.g. from the interval \, $[0,\,2\pi]$)\, for obtaining all points of the perimetre. 67 | 68 | In the analytic geometry, one speaks of the \emph{parametre of parabola} (a.k.a. \emph{latus rectum}):\, it means the chord of the parabola which is perpendicular to the axis and goes through the focus; it is the quantity $2p$ in the standard equation \, $x^2 = 2py$\; of the parabola ($p$ is the distance of the focus and the directrix). 69 | %%%%% 70 | %%%%% 71 | \end{document} 72 | -------------------------------------------------------------------------------- /00A05-Percentage.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Percentage} 4 | \pmcreated{2013-03-22 16:37:56} 5 | \pmmodified{2013-03-22 16:37:56} 6 | \pmowner{CompositeFan}{12809} 7 | \pmmodifier{CompositeFan}{12809} 8 | \pmtitle{percentage} 9 | \pmrecord{5}{38833} 10 | \pmprivacy{1} 11 | \pmauthor{CompositeFan}{12809} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | A {\em percentage} is a ratio expressed in terms of a unit being 100. A percentage is usually denoted by the symbol ``\%.'' For example, $20\%$ of $\$700.47$ is $\$175.12$ (using fixed point arithmetic to two decimal places for display). 42 | 43 | On most calculators, one sure way to calculate a percentage is by entering a decimal point before the desired percentage and multiplying that by the amount one wishes to calculate the percentage of. Some calculators have a percentage key. 44 | 45 | When tipping at most restaurants in the United States, it is customary to tip 15\% of the check to the waiter for parties of as much as four people. One common shortcut is to divide by 10 (by moving the decimal point to the left) and then add half of that amount. 46 | 47 | Note that the percentage symbol \% (Shift-5 in most American keyboard layouts) is overloaded in \TeX{} as a comment start indicator and in Mathematica as a shortcut for referring to the previous output. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A05-Proof.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Proof} 4 | \pmcreated{2013-03-22 18:10:22} 5 | \pmmodified{2013-03-22 18:10:22} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{proof} 9 | \pmrecord{4}{40736} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | A {\em proof} is an argument designed to show that a statement (usually a theorem) is true. 42 | 43 | The mathematical concept of proof differs from the scientific concept in that in mathematics, a proof is logically deduced from axioms or from other theorems which have also been logically deduced, whereas for a scientific proof a preponderance of evidence is sufficient. Thus, a valid mathematical proof assures there are no counterexamples to the proven statement. 44 | 45 | There are several kinds of proofs, one commonly used one being proof by contradiction. A proof by contradiction starts by assuming that the opposite of the theorem is true, and then proceeds to work out the consequences of that assumption until encountering a contradiction, thus proving the theorem. 46 | 47 | According to Paul Nahin, the most famous proof by contradiction is Euclid's proof of the infinitude of primes, which starts by assuming that there is in fact a largest prime number (and thus the primes are finite). Proofs that a given number is irrational (such as $\pi$ or $\sqrt{5}$) also tend to prove the irrationality of the number by at first assuming that the number is in fact rational and that there are two integers which form a ratio for the given number. 48 | 49 | Another kind of proof is the proof by induction, which starts by showing the statement is true for a small case (such as $n = 1$ when dealing with integers) and that the statement is true for a larger case when it is true for the immediately smaller case (e.g., that if it's true for $n$ it is also true for $n + 1$). Thus, showing that it is true for the small case proves that it is also true for the next larger case, and the next larger case after that, and therefore all the larger cases. 50 | 51 | A proof by construction shows that a specified object actually exists by showing how to construct that object. For example, to prove that it is possible to draw by compass and straightedge an isosceles triangle with an angle that is half of any of the two other angles, a constructive proof would give the instructions on how to draw such a triangle. 52 | 53 | \begin{thebibliography}{3} 54 | \bibitem{pn} Paul J. Nahin, {\it Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills}. Princeton: Princeton University Press (2006): 8 55 | \bibitem{tw} Thomas A. Whitelaw, {\it Introduction to Abstract Algebra}. New York: CRC Press (1995): 11 56 | \end{thebibliography} 57 | %%%%% 58 | %%%%% 59 | \end{document} 60 | -------------------------------------------------------------------------------- /00A05-RydiasMathemagicMinute.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{RydiasMathemagicMinute} 4 | \pmcreated{2013-03-22 18:57:16} 5 | \pmmodified{2013-03-22 18:57:16} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Rydia's Mathemagic Minute} 9 | \pmrecord{5}{41811} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | {\em Rydia's Mathemagic Minute} is a mini-game from {\it Final Fantasy XIII} in which Rydia presents the player a riddle in the form of four numbers, each in the range $-1 < n < 10$ which the player must then employ in conjunction with a selection of basic arithmetic operations (addition, subtraction, multiplication, division) in order to obtain 10 as a result. For example, given \{8, 8, 6, 4\}, one possible solution is $8 \times 8 - 4 \div 6$. If the player can solve the riddle, he is given another riddle of the same form, until exhausting the time limit of 90 seconds. A player may skip a riddle and move on to another one, but in so doing incurs a penalty in negative points. Correctly answering five riddles in a row results in the player receives a time limit extension. 42 | 43 | Concatenation is not allowed. For example, given \{1, 7, 2, 9\}, $19 - (2 + 7)$ is not a valid answer; the program in fact will not allow the player to even try concatenation. Answers involving negative numbers at intermediate steps are not valid either, though when they occur they're displayed in blue (in lieu of a minus sign). Fractions are not at all allowed at intermediate steps, and when the player presents an operation that would result in such a result (e.g., dividing a number by a smaller coprime number), the computer simply rejects the operation. 44 | 45 | The mini-game can be played either as a side quest on the full game, or by itself online at {\PMlinkescapetext Square Enix Members}. 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /00A05-Singleton.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Singleton} 4 | \pmcreated{2013-03-22 13:13:41} 5 | \pmmodified{2013-03-22 13:13:41} 6 | \pmowner{Koro}{127} 7 | \pmmodifier{Koro}{127} 8 | \pmtitle{singleton} 9 | \pmrecord{6}{33696} 10 | \pmprivacy{1} 11 | \pmauthor{Koro}{127} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmrelated{ATopologicalSpaceIsT_1IfAndOnlyIfEverySingletonIsClosed} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | A \textit{singleton} is a set containing a single element. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /00A05-StableSubspace.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{StableSubspace} 4 | \pmcreated{2013-03-22 17:56:52} 5 | \pmmodified{2013-03-22 17:56:52} 6 | \pmowner{lalberti}{18937} 7 | \pmmodifier{lalberti}{18937} 8 | \pmtitle{stable subspace} 9 | \pmrecord{6}{40446} 10 | \pmprivacy{1} 11 | \pmauthor{lalberti}{18937} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{invariant subspace}{StableSubspace} 16 | \pmsynonym{stable subset}{StableSubspace} 17 | \pmsynonym{invariant subset}{StableSubspace} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \begin{document} 44 | A subset $S$ of a larger set $T$ is said to a \emph{stable subset} for a function $f: T\to T$ \underline{iff} $f(S)\subset S$.\\ 45 | Alternative phrasings with the same meaning are: 46 | \begin{itemize} 47 | \item $f$ is an invariant subset for $f$ 48 | \item $f$ stabilizes $S$ 49 | \item $S$ is stable under (the action of) $f$ 50 | \item $S$ is invariant under (the action of) $f$ 51 | \item $S$ is left stable by/under $f$ 52 | \item $S$ is left invariant by/under $f$ 53 | \end{itemize} 54 | %%%%% 55 | %%%%% 56 | \end{document} 57 | -------------------------------------------------------------------------------- /00A05-Subsequence.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Subsequence} 4 | \pmcreated{2013-03-22 12:56:34} 5 | \pmmodified{2013-03-22 12:56:34} 6 | \pmowner{alozano}{2414} 7 | \pmmodifier{alozano}{2414} 8 | \pmtitle{subsequence} 9 | \pmrecord{6}{33300} 10 | \pmprivacy{1} 11 | \pmauthor{alozano}{2414} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsthm} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \newtheorem{thm}{Theorem} 42 | \newtheorem*{defn}{Definition} 43 | \newtheorem{prop}{Proposition} 44 | \newtheorem{lemma}{Lemma} 45 | \newtheorem{cor}{Corollary} 46 | 47 | \theoremstyle{definition} 48 | \newtheorem*{exa}{Example} 49 | 50 | % Some sets 51 | \newcommand{\Nats}{\mathbb{N}} 52 | \newcommand{\Ints}{\mathbb{Z}} 53 | \newcommand{\Reals}{\mathbb{R}} 54 | \newcommand{\Complex}{\mathbb{C}} 55 | \newcommand{\Rats}{\mathbb{Q}} 56 | \newcommand{\Gal}{\operatorname{Gal}} 57 | \newcommand{\Cl}{\operatorname{Cl}} 58 | \begin{document} 59 | Given a sequence $\{x_n\}_{n\in \Nats}$, any infinite subset of the sequence forms a subsequence. We formalize this as follows: 60 | 61 | \begin{defn} 62 | If $X$ is a set and $\{a_n\}_{n \in \mathbb{N}}$ is a sequence in $X$, then a \emph{subsequence} of $\{a_n\}$ is a sequence of the form $\{a_{n_r}\}_{r \in \mathbb{N}}$ where $\{n_r\}_{r \in \mathbb{N}}$ is a strictly increasing sequence of natural numbers. 63 | \end{defn} 64 | 65 | Equivalently, $\{y_n\}_{n\in \Nats}$ is a subsequence of $\{x_n\}_{n\in \Nats}$ if 66 | \begin{enumerate} 67 | \item $\{y_n\}_{n\in\Nats}$ is a sequence of elements of $X$, and 68 | 69 | \item there is a strictly increasing function $a:\Nats \to \Nats$ such that $$y_n = x_{a(n)} \quad \text{ for all } n\in\Nats.$$ 70 | \end{enumerate} 71 | 72 | \begin{exa} 73 | Let $X=\Reals$ and let $\{x_n\}$ be the sequence 74 | $$\left\{\frac{1}{n}\right\}_{n\in\Nats}=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\right\}.$$ Then, the sequence $$\{y_n\}_{n\in\Nats}=\left\{\frac{1}{n^2}\right\}_{n\in\Nats}=\left\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\ldots\right\}$$ 75 | is a subsequence of $\{x_n\}$. The subsequence of natural numbers mentioned in the definition is $\{n^2\}_{n\in\Nats}$ and the function $a:\Nats\to\Nats$ mentioned above is $a(n)=n^2$. 76 | \end{exa} 77 | %%%%% 78 | %%%%% 79 | \end{document} 80 | -------------------------------------------------------------------------------- /00A05-SumOfOddNumbers.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SumOfOddNumbers} 4 | \pmcreated{2013-03-22 14:38:35} 5 | \pmmodified{2013-03-22 14:38:35} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{sum of odd numbers} 9 | \pmrecord{15}{36230} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{11B25} 16 | \pmrelated{NumberOdd} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | The sum of the first $n$ positive odd integers can be calculated by using the well-known \PMlinkescapetext{property} of the arithmetic progression, that the sum of its \PMlinkescapetext{terms} is equal to the arithmetic mean of the first and the last \PMlinkescapetext{term}, multiplied by the number of the \PMlinkescapetext{terms}: 43 | $$\underbrace{1+3+5+7+9+\cdots+(2n\!-\!1)}_{n} = 44 | n\cdot\frac{1\!+\!(2n\!-\!1)}{2} = n^2$$ 45 | Thus, the sum of the first $n$ odd numbers is $n^2$ (this result has been proved first time in 1575 by Francesco Maurolico). 46 | 47 | Below, the odd numbers have been set to form a triangle, each $n^{\rm{th}}$ row containing the next $n$ consecutive odd numbers.\, The arithmetic mean on the row is $n^2$ and the sum of its numbers is\, $n\cdot n^2 = n^3$. 48 | \begin{eqnarray*} 49 | \begin{array}{cccccccccccccccccc} 50 | & & & & & & & & & 1 & & & & & & & &\\ 51 | & & & & & & & & 3 & & 5 & & & & & & &\\ 52 | & & & & & & & 7 & & 9 & & 11 & & & & & &\\ 53 | & & & & & & 13 & & 15 & & 17 & & 19 & & & & &\\ 54 | & & & & & 21 & & 23 & & 25 & & 27 & & 29 & & & &\\ 55 | & & & & 31 & & 33 & &35 & &37 & &39 & & 41 & & &\\ 56 | & & & & &\vdots & & & & \vdots & & & & \vdots& & & & \\ 57 | \end{array} 58 | \end{eqnarray*} 59 | %%%%% 60 | %%%%% 61 | \end{document} 62 | -------------------------------------------------------------------------------- /00A05-TimeInvariant.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TimeInvariant} 4 | \pmcreated{2013-03-22 15:02:14} 5 | \pmmodified{2013-03-22 15:02:14} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{time invariant} 9 | \pmrecord{5}{36748} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmrelated{AutonomousSystem} 16 | \pmdefines{time-invariant} 17 | \pmdefines{shift-invariant} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | A dynamical system is {\bf time-invariant} if its generating formula is dependent on state only, and independent of time. A synonym for time-invariant is autonomous. The complement of time-invariant is time-varying (or nonautonomous). 44 | 45 | For example, the continuous-time system $\dot{x}=f(x,t)$ is time-invariant if and only if $f(x,t_1)\equiv f(x,t_2)$ for all valid states $x$ and times $t_1$ and $t_2$. Thus $\dot{x}=\sin x$ is time-invariant, while $\dot{x}=\frac{\sin x}{1+t}$ is time-varying. 46 | 47 | Likewise, the discrete-time system $x[n]=f[x,n]$ is time-invariant (also called shift-invariant) if and only if $f[x,n_1]\equiv f[x,n_2]$ for all valid states $x$ and time indices $n_1$ and $n_2$. Thus $x[n]=2 x[n-1]$ is time-invariant, while $x[n]=2 n x[n-1]$ is time-varying. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A05-Welldefined.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Welldefined} 4 | \pmcreated{2013-03-22 17:31:32} 5 | \pmmodified{2013-03-22 17:31:32} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{well-defined} 9 | \pmrecord{9}{39921} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmsynonym{well defined}{Welldefined} 16 | \pmrelated{function} 17 | \pmrelated{WellDefinednessOfProductOfFinitelyGeneratedIdeals} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | \usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \theoremstyle{definition} 44 | \newtheorem*{thmplain}{Theorem} 45 | 46 | \begin{document} 47 | A mathematical concept is \emph{well-defined} (German {\em wohldefiniert}, French {\em bien d\'efini}), if its contents is 48 | \PMlinkescapetext{independent} on the form or the alternative representative which is used for defining it. 49 | 50 | For example, in defining the \PMlinkname{power}{FractionPower} $x^r$ with $x$ a positive real and $r$ a rational number, 51 | we can freely choose the fraction form $\frac{m}{n}$ ($m\in\mathbb{Z}$,\, $n\in\mathbb{Z}_+$) of $r$ and take 52 | $$x^r \;:=\; \sqrt[n]{x^m}$$ 53 | and be sure that the value of $x^r$ does not depend on that choice (this is justified in the entry fraction power). So, 54 | the $x^r$ is well-defined. 55 | 56 | In many instances well-defined is a synonym for the formal definition of a function between sets. For example, 57 | the function\, $f(x) := x^2$\, is a well-defined function from the real numbers to the real numbers because 58 | every input, $x$, is assigned to precisely one output, $x^2$. However,\, $f(x) := \pm\sqrt{x}$\, is not well-defined 59 | in that one input $x$ can be assigned any one of two possible outputs, $\sqrt{x}$ or $-\sqrt{x}$. 60 | 61 | More subtle examples include expressions such as 62 | \begin{equation*} 63 | f\!\left(\frac{a}{b}\right) \;:=\; a\!+\!b,\quad \frac{a}{b}\in\mathbb{Q}. 64 | \end{equation*} 65 | Certainly every input has an output, for instance,\, $f(1/2) = 3$. However, the expression is \emph{not} 66 | well-defined since\, $1/2 = 2/4$\, yet\, $f(1/2) = 3$\, while\, $f(2/4) = 6$\, and\, $3 \neq 6$. 67 | 68 | One must question whether a function is well-defined whenever it is defined on a domain of equivalence classes 69 | in such a manner that each output is determined for a representative of each equivalence class. For example, the 70 | function\, $f(a/b) := a\!+\!b$\, was defined using the representative $a/b$ of the equivalence class of fractions 71 | equivalent to $a/b$. 72 | %%%%% 73 | %%%%% 74 | \end{document} 75 | -------------------------------------------------------------------------------- /00A05-WindowsCalculator.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{WindowsCalculator} 4 | \pmcreated{2013-03-22 16:39:22} 5 | \pmmodified{2013-03-22 16:39:22} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Windows Calculator} 9 | \pmrecord{8}{38861} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A05} 15 | \pmclassification{msc}{01A07} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | The {\em Windows Calculator} is a software calculator that comes bundled with the Windows operating system. The basic mode is called ``Standard'' and is the default, Scientific mode has most of the operations available on a typical scientific calculator. Note that switching between modes causes the loss of the current value displayed (unless of course that value is 0). For some reason, Standard mode has a square root key but Scientific mode does not. As a workaround in scientific mode, one can enter, say, \verb=[2] [x^y] [0] [.] [5]=. 43 | 44 | Division by zero causes an error condition that must be cleared with the C key on the displayed keyboard (or the Escape key on the computer's keyboard). Integer values smaller than $10^{32}$ can be displayed in all their digits. According to the Help, the Windows Calculator truncates $\pi$ to 32 digits, but rational numbers are stored internally ``as fractions''. 45 | 46 | Like most scientific calculators, the Windows Calculator can display results in binary, octal and hexadecimal, but is limited to integers in those bases. Additionally, negative numbers are shown in two's complement (and the sign change key performs two's complement on the displayed value). In those bases, the user can choose the data size: quadruple word (the default), double word, word or byte. Overflows don't trigger any kind of exception or error notification, the calculator quietly discards the more significant digits and displays the least significant digits that will fit in the currently selected data size. 47 | 48 | Like the Mac OS Calculator, for the Windows Calculator $0^0 = 1$. 49 | 50 | \begin{thebibliography}{1} 51 | \bibitem{dk} David A. Karp, Tim O'Reilly \& Troy Mott, {\it Windows XP in a Nutshell} Cambridge: O'Reilly (2002): 114 - 117 52 | \end{thebibliography} 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /00A06-Multiplication.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Multiplication} 4 | \pmcreated{2013-03-22 16:35:37} 5 | \pmmodified{2013-03-22 16:35:37} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{multiplication} 9 | \pmrecord{10}{38789} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{11B25} 16 | \pmclassification{msc}{00A05} 17 | \pmrelated{Product} 18 | \pmrelated{ProductOfNegativeNumbers} 19 | \pmrelated{FactorsWithMinusSign} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | 45 | \begin{document} 46 | {\em Multiplication} is a mathematical operation in which two or more numbers are added up to themselves by a factor of other numbers. For example, $2 \times 3 = 2 + 2 + 2 = 3 + 3 = 6$. The numbers may be real, imaginary or complex, they may be integers or fractions. Among real numbers, if an odd number of multiplicands are negative, the overall result is negative; if an even number of multiplicands are negative, the overall result is positive. Two examples: $(-3) \times (-5) = 15$; $(-2) \times (-3) \times (-5) = (-30)$. 47 | 48 | The usual operator is the cross with its four arms of equal length pointing northeast, northwest, southeast and southwest: $\times$. Other options are the central dot $\cdot$ and the tacit multiplication operator. In many computer programming languages the asterisk is often used as it is almost always available on the keyboard (Shift-8 in most American layouts, as well as dedicated key if the keyboard has a numeric keypad), and this is the operator likely to be used in a computer implementation of a reverse Polish notation calculator. In Mathematica, the space can sometimes function as a multiplication operator, but more experienced users warn novices not to rely on this feature. 49 | 50 | Just as with addition, multiplication is commutative: $xyz = xzy = yxz$, etc. 51 | 52 | The iterative operator is the Greek capital letter pi: $$\prod_{i = 1}^n a_i,$$ which is a compact way of writing $a_1 \times a_2 \times \ldots \times a_n$. 53 | 54 | Multiplication of complex numbers is helped by the following identity: $(a + bi) \times (x + yi)=(ax - by) + (ay + bx)i$. To give three examples: $(17 + 29i)(11 + 38i) = -915 + 965i$ (the result has both real and imaginary parts), $(1 + 2i)(1 - 2i) = 5$ (the result is a real prime) and $(4 + 7i)(7 + 4i) = 65i$ (the result has only an imaginary part). 55 | %%%%% 56 | %%%%% 57 | \end{document} 58 | -------------------------------------------------------------------------------- /00A06-PlusSign.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PlusSign} 4 | \pmcreated{2013-03-22 17:35:43} 5 | \pmmodified{2013-03-22 17:35:43} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{plus sign} 9 | \pmrecord{5}{40009} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{00A05} 16 | \pmsynonym{plus}{PlusSign} 17 | \pmrelated{Sum} 18 | \pmrelated{SumOfSeries} 19 | \pmrelated{SignumFunction} 20 | \pmrelated{OppositeNumber} 21 | \pmrelated{ProductOfNegativeNumbers} 22 | 23 | \endmetadata 24 | 25 | % this is the default PlanetMath preamble. as your knowledge 26 | % of TeX increases, you will probably want to edit this, but 27 | % it should be fine as is for beginners. 28 | 29 | % almost certainly you want these 30 | \usepackage{amssymb} 31 | \usepackage{amsmath} 32 | \usepackage{amsfonts} 33 | 34 | % used for TeXing text within eps files 35 | %\usepackage{psfrag} 36 | % need this for including graphics (\includegraphics) 37 | %\usepackage{graphicx} 38 | % for neatly defining theorems and propositions 39 | \usepackage{amsthm} 40 | % making logically defined graphics 41 | %%%\usepackage{xypic} 42 | 43 | % there are many more packages, add them here as you need them 44 | 45 | % define commands here 46 | 47 | \theoremstyle{definition} 48 | \newtheorem*{thmplain}{Theorem} 49 | 50 | \begin{document} 51 | There are two main uses of the {\em plus sign} ``$+$'' (which is a simplified form of ``\&'') in the mathematics and the applying sciences: 52 | \begin{itemize} 53 | \item The original use is as the sign for the binary operation {\em addition} of numbers and other elements of rings and algebras, vectors, etc.: 54 | $$a\!+\!b\; := \mbox{\, the sum of\, } a \mbox{\, and\, }b$$ 55 | \item There is also a special use for the unary operation {\em identity mapping} concerning numbers and other ring elements: 56 | $$+a\; :=\; a \mbox{\; (for all } a)$$ 57 | \end{itemize} 58 | %%%%% 59 | %%%%% 60 | \end{document} 61 | -------------------------------------------------------------------------------- /00A06-SimpleInterest.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SimpleInterest} 4 | \pmcreated{2013-03-22 16:40:06} 5 | \pmmodified{2013-03-22 16:40:06} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{simple interest} 9 | \pmrecord{7}{38875} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{00A69} 16 | \pmclassification{msc}{91B28} 17 | \pmrelated{CompoundInterest} 18 | \pmrelated{InterestRate} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb,amscd} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \usepackage{tabls} 26 | % used for TeXing text within eps files 27 | %\usepackage{psfrag} 28 | % need this for including graphics (\includegraphics) 29 | %\usepackage{graphicx} 30 | % for neatly defining theorems and propositions 31 | %\usepackage{amsthm} 32 | % making logically defined graphics 33 | %%\usepackage{xypic} 34 | \usepackage{pst-plot} 35 | \usepackage{psfrag} 36 | 37 | % define commands here 38 | 39 | \begin{document} 40 | Suppose a bank account is opened at time $0$ and $M_0$ is deposited into the account. A \emph{simple interest} is interest with the following characteristics: 41 | \begin{enumerate} 42 | \item it is earned at subsequent time periods $t,2t,\ldots$, where $t$ is the length of the initial time interval (1 for 1 month, 12 for 1 year, etc...) 43 | \item the interest earned at the end of each time period is the same regardless of the time period 44 | \end{enumerate} 45 | The following table illustrates the structure of the simple interest. 46 | 47 | \begin{center} 48 | \begin{tabular}{|c||c|c|c|} 49 | \hline time period at & principal & interest & interest accrued \\ 50 | \hline\hline $0$ & $M_0$ & $0$ & $0$ \\ 51 | \hline $t$ & $M_0$ & $i$ & $i$ \\ 52 | \hline $2t$ & $M_0$ & $i$ & $2i$ \\ 53 | \hline $3t$ & $M_0$ & $i$ & $3i$ \\ 54 | \hline $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\ 55 | \hline $nt$ & $M_0$ & $i$ & $ni$ \\ 56 | \hline 57 | \end{tabular} 58 | \end{center} 59 | 60 | The ``total'' interest $i(nt)$ earned (accrued) at the end of time $nt$ is $ni$. If the account is closed and the money withdrawn at the end of $nt$, and the total amount of money received is $$M(nt)=M_0+ni.$$ 61 | 62 | The interest rate associated with the simple interest as presented above between two time periods, say $at$ and $bt$, is given by 63 | $$r(at,bt)=\frac{1}{M_0}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_0t},$$ 64 | which does not depend on the choice of $a$ and $b$. In other words, the original principal $M_0$, the amount of interest $i$, and the length of the initial time interval $t$ are enough to determine the interest rate. 65 | 66 | \textbf{Remark}. 67 | \begin{itemize} 68 | \item 69 | The expression for the effective interest rate for simple interest is a bit more complicated: 70 | $$\operatorname{eff.}r(at,bt)=\frac{1}{M(at)}\frac{i(bt)-i(at)}{bt-at}= 71 | \frac{1}{M_0+ai}\frac{i}{t},$$ 72 | which decreases with increasing $a$. Imagine as $a$ becomes very large, the increase in interest has practically no impact on the ``accumulated'' principal $M(at)$. 73 | \item 74 | More generally, we say that an interest is \emph{simple} if its interest rate $r$ is constant with respect to time $t$. Solving 75 | $$r=\frac{1}{M_0}\frac{i(t)-i(0)}{t-0}$$ 76 | for $i(t)$, we get $i(t)=M_0rt$, or that the accrued interest is a linear function of $t$. It grows directly proportionally with respect to time. 77 | \end{itemize} 78 | %%%%% 79 | %%%%% 80 | \end{document} 81 | -------------------------------------------------------------------------------- /00A06-Subtraction.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Subtraction} 4 | \pmcreated{2013-03-22 16:35:31} 5 | \pmmodified{2013-03-22 16:35:31} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{subtraction} 9 | \pmrecord{6}{38787} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{00A05} 16 | \pmclassification{msc}{11B25} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | 42 | \begin{document} 43 | {\em Subtraction} is a mathematical operation in which the value of a number is decreased by the values of one or more other numbers. Subtraction can be seen as a kind of addition with negative numbers. For example, $7 - 4 = 7 + (-4) = 3$. 44 | 45 | The usual operator looks like a dash: $-$. This operator is used in standard infix notation as well as in Polish notation and reverse Polish notation. 46 | 47 | Besides the possibility of overflow or underflow, subtraction presents no problems for fixed point arithmetic provided the operands are representable in fixed point to begin with. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A06-TableOfMultiplicationUpTo12.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TableOfMultiplicationUpTo12} 4 | \pmcreated{2013-03-22 16:36:14} 5 | \pmmodified{2013-03-22 16:36:14} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{table of multiplication up to 12} 9 | \pmrecord{6}{38799} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Data Structure} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{00A05} 16 | \pmclassification{msc}{11B25} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | 42 | \begin{document} 43 | Because of the commutative property of multiplication, it does not matter if the row or the column gives the first operand. 44 | 45 | \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|} 46 | $\times$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 47 | 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 48 | 2 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 \\ 49 | 3 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27 & 30 & 33 & 36 \\ 50 | 4 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & 44 & 48 \\ 51 | 5 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 55 & 60 \\ 52 | 6 & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 & 60 & 66 & 72 \\ 53 | 7 & 7 & 14 & 21 & 28 & 35 & 42 & 49 & 56 & 63 & 70 & 77 & 84 \\ 54 | 8 & 8 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 & 80 & 88 & 96 \\ 55 | 9 & 9 & 18 & 27 & 36 & 45 & 54 & 63 & 72 & 81 & 90 & 99 & 108 \\ 56 | 10 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 & 110 & 120 \\ 57 | 11 & 11 & 22 & 33 & 44 & 55 & 66 & 77 & 88 & 99 & 110 & 121 & 132 \\ 58 | 12 & 12 & 24 & 36 & 48 & 60 & 72 & 84 & 96 & 108 & 120 & 132 & 144 \\ 59 | \end{tabular} 60 | 61 | Obviously the longest northwest to southeast diagonal contains numbers of the form $n^2$. 62 | %%%%% 63 | %%%%% 64 | \end{document} 65 | -------------------------------------------------------------------------------- /00A06-ThereforeSign.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ThereforeSign} 4 | \pmcreated{2013-03-22 17:55:51} 5 | \pmmodified{2013-03-22 17:55:51} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{therefore sign} 9 | \pmrecord{8}{40425} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A06} 15 | \pmclassification{msc}{00A05} 16 | \pmrelated{RingsOfRationalNumbers} 17 | \pmrelated{LogarithmicScale} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | \usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \theoremstyle{definition} 44 | \newtheorem*{thmplain}{Theorem} 45 | 46 | \begin{document} 47 | The {\em therefore sign} ``$\therefore$'' is used especially in handwritten mathematical text as a shorthand of the \PMlinkescapetext{word `therefore' or `thus', between some sentences} or relations: 48 | $$S_1 \quad \therefore \; S_2$$ 49 | It expresses that $S_2$ has been inferred from $S_1$ or from $S_1$ and some preceding facts. The sign is rather a punctuation mark than a symbol of logical implication. Grammatically, it could be characterised a conclusive coordinating \PMlinkescapetext{conjunction}. The usage of the symbol is not mathematically well-defined, and it often means `we can conclude in context' or `we can conclude from statements already shown or assumed to be true'.\\ 50 | 51 | For example, in determining an angle of a right triangle, one may write 52 | $$\sin\alpha = \frac{1}{2} \quad \therefore \; \alpha = 30^\circ$$ 53 | Here, ``$\therefore$'' does not \PMlinkescapetext{represent} a proper implication ``$\Rightarrow$'', since the exact implication here would be 54 | $$\sin\alpha = \frac{1}{2} \;\, \Leftarrow \;\, \alpha = 30^\circ.$$ 55 | To obtain a strict implication, we would need to introduce some of the context. 56 | For instance, we know that, since $\alpha$ is an angle of a right triangle, 57 | $0^\circ \le \alpha \le 90^\circ$, 58 | so what we wrote could be interpreted as the implication 59 | $$\sin\alpha = \frac{1}{2} \; \land \; 60 | 0^\circ \le \alpha \le 90^\circ 61 | \;\, \Rightarrow \;\, \alpha = 30^\circ.$$ 62 | %%%%% 63 | %%%%% 64 | \end{document} 65 | -------------------------------------------------------------------------------- /00A07-NesbittsInequality.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{NesbittsInequality} 4 | \pmcreated{2013-03-22 12:36:59} 5 | \pmmodified{2013-03-22 12:36:59} 6 | \pmowner{mathwizard}{128} 7 | \pmmodifier{mathwizard}{128} 8 | \pmtitle{Nesbitt's inequality} 9 | \pmrecord{7}{32875} 10 | \pmprivacy{1} 11 | \pmauthor{mathwizard}{128} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A07} 15 | \pmrelated{ShapiroInequality} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | Nesbitt's inequality says, that for positive real $a$, $b$ and $c$ we have: 42 | $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.$$ 43 | This is a special case of Shapiro's inequality. 44 | %%%%% 45 | %%%%% 46 | \end{document} 47 | -------------------------------------------------------------------------------- /00A07-ProofOfNesbittsInequality.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ProofOfNesbittsInequality} 4 | \pmcreated{2013-03-22 12:37:01} 5 | \pmmodified{2013-03-22 12:37:01} 6 | \pmowner{mathwizard}{128} 7 | \pmmodifier{mathwizard}{128} 8 | \pmtitle{proof of Nesbitt's inequality} 9 | \pmrecord{6}{32876} 10 | \pmprivacy{1} 11 | \pmauthor{mathwizard}{128} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A07} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | Starting from Nesbitt's inequality 41 | $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}$$ 42 | we transform the left hand side: 43 | $$\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\geq\frac{3}{2}.$$ 44 | Now this can be transformed into: 45 | $$((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\geq 9.$$ 46 | Division by 3 and the right \PMlinkescapetext{factor} yields: 47 | $$\frac{(a+b)+(a+c)+(b+c)}{3}\geq\frac{3}{\frac{1}{a+b}+\frac{1}{a+c}+ \frac{1}{b+c}}.$$ 48 | Now on the left we have the arithmetic mean and on the right the harmonic mean, so this inequality is true. 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /00A08-CabtaxiNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CabtaxiNumber} 4 | \pmcreated{2013-03-22 17:56:08} 5 | \pmmodified{2013-03-22 17:56:08} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{cabtaxi number} 9 | \pmrecord{4}{40432} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A08} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | A {\em cabtaxi number} for a given $n$ is the smallest positive number which can be written as $a^3 + b^3$ in $n$ different ways, with either $a$ or $b$ allowed to be negative integers. For example, 91 is the 2nd cabtaxi number since it can be expressed $(-5)^3 + 6^3 = 3^3 + 4^3 = 91$. The known cabtaxi numbers are 1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, listed in A047696 of Sloane's OEIS. Adding the restriction $a \geq b > 0$ gives the definition for the taxicab numbers. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /00A08-LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind} 4 | \pmcreated{2013-03-22 16:04:14} 5 | \pmmodified{2013-03-22 16:04:14} 6 | \pmowner{Mravinci}{12996} 7 | \pmmodifier{Mravinci}{12996} 8 | \pmtitle{large integers that are or might be the smallest of their kind} 9 | \pmrecord{15}{38126} 10 | \pmprivacy{1} 11 | \pmauthor{Mravinci}{12996} 12 | \pmtype{Feature} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A08} 15 | \pmrelated{SmallIntegersThatAreOrMightBeTheLargestOfTheirKind} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | \PMlinkescapeword{class} 43 | \PMlinkescapeword{length} 44 | 45 | For the purpose of this feature, the arbitrary cutoff is $10^7$. 46 | 47 | {\bf 19099919} is the smallest prime to start a Cunningham chain of length 8. 48 | 49 | {\bf 85864769} is the smallest prime to start a Cunningham chain of length 9. 50 | 51 | {\bf 545587687} is the smallest class 13+ prime in the Erdos-Selfridge classification of primes. 52 | 53 | {\bf 635318657} is the smallest number that can be expressed as a sum of two fourth powers in two different ways. 54 | 55 | {\bf 823766851} is the smallest prime with primitive root 48. 56 | 57 | {\bf 906150257} is the smallest counterexample to P\'olya's conjecture. 58 | 59 | {\bf 1023456789} is the smallest pandigital number in base 10. 60 | 61 | {\bf 1704961513} is the smallest class 14+ prime in the Erd\H{o}s-Selfridge classification of primes. 62 | 63 | {\bf 10123457689} is the smallest pandigital prime in base 10. 64 | 65 | {\bf 26089808579} is the smallest prime to start a Cunningham chain of length 10. 66 | 67 | {\bf 665043081119} is the smallest prime to start a Cunningham chain of length 11. 68 | 69 | {\bf 554688278429} is the smallest prime to start a Cunningham chain of length 12. 70 | 71 | $10^{13} + 1$ is, as of 2005, the smallest candidate for a counterexample to the Mertens conjecture (though the smallest counterexample could turn out to be as large as $3.21 \times 10^{64}$). 72 | 73 | {\bf 4090932431513069} is the smallest prime to start a Cunningham chain of length 13. 74 | 75 | {\bf 95405042230542329} is the smallest prime to start a Cunningham chain of length 14. 76 | 77 | {\bf 810433818265726529159} is the smallest prime known to start a Cunningham chain of length 16, but there could be a smaller such prime. 78 | 79 | {\bf 439351292910452432574786963588089477522344721} is the smallest prime in Paul Hoffman's erroneous version of Wilf's primefree sequence in which $a_1 = 3794765361567513$, $a_2 = 20615674205555510$ and $a_n = a_{n - 2} + a_{n - 1}$ for $n > 2$. 80 | 81 | If an odd perfect number exists, it is at least $10^{300} + 1$. 82 | %%%%% 83 | %%%%% 84 | \end{document} 85 | -------------------------------------------------------------------------------- /00A08-TaxicabNumbers.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TaxicabNumbers} 4 | \pmcreated{2013-03-22 15:43:00} 5 | \pmmodified{2013-03-22 15:43:00} 6 | \pmowner{alozano}{2414} 7 | \pmmodifier{alozano}{2414} 8 | \pmtitle{taxicab numbers} 9 | \pmrecord{6}{37664} 10 | \pmprivacy{1} 11 | \pmauthor{alozano}{2414} 12 | \pmtype{Feature} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A08} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsthm} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \newtheorem{thm}{Theorem} 42 | \newtheorem{defn}{Definition} 43 | \newtheorem{prop}{Proposition} 44 | \newtheorem{lemma}{Lemma} 45 | \newtheorem{cor}{Corollary} 46 | 47 | \theoremstyle{definition} 48 | \newtheorem{exa}{Example} 49 | 50 | % Some sets 51 | \newcommand{\Nats}{\mathbb{N}} 52 | \newcommand{\Ints}{\mathbb{Z}} 53 | \newcommand{\Reals}{\mathbb{R}} 54 | \newcommand{\Complex}{\mathbb{C}} 55 | \newcommand{\Rats}{\mathbb{Q}} 56 | \newcommand{\Gal}{\operatorname{Gal}} 57 | \newcommand{\Cl}{\operatorname{Cl}} 58 | \begin{document} 59 | The number $1729$ has a reputation of its own. The reason is the famous exchange between \PMlinkexternal{G. H. Hardy}{http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html}, a famous British mathematician (1877-1947), and \PMlinkexternal{Srinivasa Ramanujan}{http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html} , one of India's greatest mathematical geniuses (1887-1920): 60 | 61 | \begin{quote} 62 | In 1917, during one visit to Ramanujan in a hospital (he was ill for much of his last three years), Hardy mentioned that the number of the taxi cab 63 | that had brought him was $1729$, which, as numbers go, Hardy thought 64 | was ``rather a dull number''. At this, Ramanujan perked up, and said 65 | ``No, it is a very interesting number; it is the smallest number 66 | expressible as a sum of two cubes in two different ways.'' 67 | \end{quote} 68 | 69 | Indeed: 70 | $$1729=1+12^3=9^3+10^3.$$ 71 | Moreover, there are other reasons why $1729$ is far from dull. $1729$ is the third Carmichael number. Even more strange, beginning at the 72 | $1729$th decimal digit of the transcental number $e$, the next ten 73 | successive digits of $e$ are 0719425863. This is the first appearance 74 | of all ten digits in a row without repititions. 75 | 76 | More generally, the smallest natural number which can be expressed as the sum of $n$ positive cubes is called the $n$th taxicab number. The first taxicab numbers are: 77 | $$2=1^3+1^3,\ 1729 =1^3+12^3=9^3+10^3,\ 87539319=167^3+436^3=228^3+423^3=255^3+414^3$$ 78 | followed by $6963472309248$ (found by E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel in 1991) and $48988659276962496$ (found by David Wilson on November 21st, 1997). 79 | %%%%% 80 | %%%%% 81 | \end{document} 82 | -------------------------------------------------------------------------------- /00A20-Canonical.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Canonical} 4 | \pmcreated{2013-03-22 14:44:32} 5 | \pmmodified{2013-03-22 14:44:32} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{canonical} 9 | \pmrecord{6}{36379} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmrelated{CanonicalFormOfElementOfNumberField} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | \usepackage{amsthm} 28 | 29 | \usepackage{mathrsfs} 30 | 31 | % used for TeXing text within eps files 32 | %\usepackage{psfrag} 33 | % need this for including graphics (\includegraphics) 34 | %\usepackage{graphicx} 35 | % for neatly defining theorems and propositions 36 | % 37 | % making logically defined graphics 38 | %%%\usepackage{xypic} 39 | 40 | % there are many more packages, add them here as you need them 41 | 42 | % define commands here 43 | 44 | \newcommand{\sR}[0]{\mathbb{R}} 45 | \newcommand{\sC}[0]{\mathbb{C}} 46 | \newcommand{\sN}[0]{\mathbb{N}} 47 | \newcommand{\sZ}[0]{\mathbb{Z}} 48 | 49 | \usepackage{bbm} 50 | \newcommand{\Z}{\mathbbmss{Z}} 51 | \newcommand{\C}{\mathbbmss{C}} 52 | \newcommand{\R}{\mathbbmss{R}} 53 | \newcommand{\Q}{\mathbbmss{Q}} 54 | 55 | 56 | 57 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 58 | \newcommand*{\abs}[1]{| #1 |} 59 | 60 | 61 | 62 | \newtheorem{thm}{Theorem} 63 | \newtheorem{defn}{Definition} 64 | \newtheorem{prop}{Proposition} 65 | \newtheorem{lemma}{Lemma} 66 | \newtheorem{cor}{Corollary} 67 | \begin{document} 68 | A mathematical object is said to be \emph{canonical} 69 | if it arises in a natural way without introducing any additional objects. 70 | 71 | \subsubsection*{Examples} 72 | \begin{enumerate} 73 | \item Suppose $A\times B$ is the Cartesian product of sets $A,B$. 74 | Then $A\times B$ has two \PMlinkescapetext{canonical projections} 75 | $A\times B\to A$ 76 | and $A\times B\to B$ defined in a natural way. Of course, if 77 | we assume more structure of $A,B$ there are also other projections. 78 | \item \PMlinkname{canonical projection}{CanonicalProjection} (in group theory) 79 | \end{enumerate} 80 | 81 | \subsubsection*{Notes} 82 | For a discussion of the theological use of canonical, see \cite{wiki}. 83 | 84 | \begin{thebibliography}{9} 85 | \bibitem{wiki} Wikipedia, article on \PMlinkexternal{canonical}{http://en.wikipedia.org/wiki/Canonical}. 86 | \end{thebibliography} 87 | %%%%% 88 | %%%%% 89 | \end{document} 90 | -------------------------------------------------------------------------------- /00A20-CompletingTheSquare.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CompletingTheSquare} 4 | \pmcreated{2013-03-22 13:36:27} 5 | \pmmodified{2013-03-22 13:36:27} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{completing the square} 9 | \pmrecord{14}{34237} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Algorithm} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmrelated{SquareOfSum} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | \newcommand{\sR}[0]{\mathbb{R}} 28 | 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | Let us consider the expression $x^2+xy$, where 44 | $x$ and $y$ are real (or complex) numbers. 45 | Using the formula 46 | $$(x+y)^2 = x^2+2xy +y^2$$ 47 | we can write 48 | \begin{eqnarray*} 49 | x^2+xy &=& x^2+xy+ 0\\ 50 | &=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\ 51 | &=& \left(x+\frac{y}{2}\right)^2-\frac{y^2}{4}. 52 | \end{eqnarray*} 53 | This manipulation is called \emph{completing the square} \cite{adams} in 54 | $x^2+xy$, or completing the square $x^2$. 55 | 56 | Replacing $y$ by $-y$, we also have 57 | $$x^2-xy = \left(x-\frac{y}{2}\right)^2-\frac{y^2}{4}.$$ 58 | 59 | Here are some applications of this method: 60 | \begin{itemize} 61 | \item 62 | \PMlinkname{Derivation of the solution formula to the quadratic equation}{DerivationOfQuadraticFormula}. 63 | \item Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle 64 | \begin{align*} 65 | x^2+y^2+2x+4y=5\Rightarrow (x+1)^2 + (y+2)^2= 10, 66 | \end{align*} 67 | from which it is frequently easier to read off important information (the center, radius, etc.) 68 | \item Completing the square can also be used to find the extremal value 69 | of a quadratic polynomial \cite{thompson} without calculus. 70 | Let us illustrate this for the polynomial $p(x)=4x^2+8x+9$. 71 | Completing the square yields 72 | \begin{eqnarray*} 73 | p(x) &=& (2x+2)^2-4 +9 \\ 74 | &=& (2x+2)^2+5 \\ 75 | &\ge & 5, 76 | \end{eqnarray*} 77 | since $(2x+2)^2\ge 0$. Here, equality holds if and 78 | only if $x=-1$. 79 | Thus $p(x)\ge 5$ for all $x\in \sR$, and $p(x)=5$ if and only if 80 | $x=-1$. 81 | It follows that $p(x)$ has a global minimum at $x=-1$, where $p(-1)=5$. 82 | \item Completing the square can also be used as an integration technique 83 | to integrate, for example the function $\displaystyle \frac{1}{4x^2+8x+9}$ \cite{adams}. 84 | \end{itemize} 85 | 86 | 87 | \begin{thebibliography}{9} 88 | \bibitem {adams} R. Adams, \emph{Calculus, a complete course}, 89 | Addison-Wesley Publishers Ltd, 3rd ed. 90 | \bibitem {thompson} 91 | \emph{Matematiklexikon} (in Swedish), 92 | J. Thompson, T. Martinsson, Wahlstr\"om \& Widstrand, 1991. 93 | \end{thebibliography} 94 | 95 | (Anyone has an English reference?) 96 | %%%%% 97 | %%%%% 98 | \end{document} 99 | -------------------------------------------------------------------------------- /00A20-Counterexample.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Counterexample} 4 | \pmcreated{2013-03-22 14:43:00} 5 | \pmmodified{2013-03-22 14:43:00} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{counterexample} 9 | \pmrecord{5}{36338} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | A \emph{counterexample} is an example which is used to prove that a statement is false. 41 | 42 | For instance, to disprove that the statement ``For all $n$, $2^{2^n} + 1$ is prime.'', one could exhibit $5$ as a counterexample; since $2^{2^5} + 1 = 641 \times 6700417$, the statement is false. 43 | %%%%% 44 | %%%%% 45 | \end{document} 46 | -------------------------------------------------------------------------------- /00A20-DigitalObject.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{DigitalObject} 4 | \pmcreated{2013-03-22 15:30:38} 5 | \pmmodified{2013-03-22 15:30:38} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{digital object} 9 | \pmrecord{11}{37374} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | A \emph{digital object} in a digital library is the textual or multimedia data and the metadata. 41 | 42 | Formally, a digital object $DO$ is a quadruple $(h, SM, ST, SS)$ where 43 | \begin{enumerate} 44 | \item $h\in H$, where $H$ is a set of universally unique handles (labels); 45 | \item $SM = \lbrace sm_1, sm_2, . . . , sm_n\rbrace$ is a set of streams; 46 | \item $ST = \lbrace st_1, st_2, . . . , st_m\rbrace$ is a set of structural metadata specifications; and 47 | \item $SS = \lbrace stsm_1, stsm_2, . . . , stsm_p\rbrace$ is a set of Structured Streams functions defined from the streams in $SM$ and from the structures in $ST$. 48 | \end{enumerate} 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /00A20-FamousOpenQuestionsInMathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{FamousOpenQuestionsInMathematics} 4 | \pmcreated{2013-03-22 14:42:52} 5 | \pmmodified{2013-03-22 14:42:52} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{famous open questions in mathematics} 9 | \pmrecord{15}{36336} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmrelated{TwoGeneratorProperty} 16 | \pmrelated{MillenniumProblems} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | Despite the fact that at least a generation of mathematicians has tried to solve these problems, they still remain open: 43 | 44 | \begin{enumerate} 45 | \item Goldbach conjecture 46 | \item Twin prime conjecture 47 | \item Poincar\'e conjecture 48 | \item Riemann Hypothesis 49 | \item Birch and Swinnerton-Dyer conjecture 50 | \item P vs NP problem 51 | \item Hodge conjecture 52 | \item Solution to the Navier-Stokes equations 53 | \item Collatz problem 54 | \item Schanuel's conjecture 55 | \item Beal conjecture 56 | \item Irrationality of Euler's constant $\gamma$ 57 | \item Existence of an odd perfect number 58 | \item Koethe conjecture 59 | \item Invariant subspace problem 60 | \item Lehmer's conjecture 61 | \end{enumerate} 62 | 63 | Open problems 3 to 8, together with the quest for a mathematical foundation explaining the mass gap property in Yang-Mills theory, constitute a collection of problems known as the Millennium Problems. Please see \PMlinkexternal{http://www.claymath.org/millennium/}{http://www.claymath.org/millennium/} for more detail. 64 | 65 | \textbf{Note:} 66 | A mathematical foundation for the partial solution of the the mass gap property in Yang-Mills theory has been published by G. Cleaver and K. Tanaka (2000): ``Ratio of Quark Masses in Duality Theories.'', 67 | \PMlinkexternal{on line}{http://arxiv.org/abs/hep-th/0002089v1}, and can be accessed as a 68 | \PMlinkexternal{PDF file}{http://arxiv.org/PS_cache/hep-th/pdf/0002/0002089v1.pdf} . 69 | 70 | %%%%% 71 | %%%%% 72 | \end{document} 73 | -------------------------------------------------------------------------------- /00A20-Obvious.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Obvious} 4 | \pmcreated{2013-03-22 14:43:42} 5 | \pmmodified{2013-03-22 14:43:42} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{obvious} 9 | \pmrecord{10}{36357} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmsynonym{easy to see}{Obvious} 16 | \pmsynonym{clear}{Obvious} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | \PMlinkescapeword{language} 43 | \PMlinkescapeword{simple} 44 | \PMlinkescapeword{cuts} 45 | \PMlinkescapeword{length} 46 | \PMlinkescapeword{potential} 47 | \PMlinkescapeword{terms} 48 | \PMlinkescapeword{average} 49 | 50 | 51 | Mathematicians use phrases like ``it is obvious that'', ``it is easy to see that'', ``it is clear that'', and ``is trivial'' to indicate that some steps have been omitted. The use of such language may be classified under three headings --- honest, dishonest, and pedagogical. 52 | 53 | The honest use of these phrases occurs when only a few steps have been omitted and these steps are simple enough that the average reader can easily fill in the gaps. Omitting such steps can be beneficial because it cuts down the length of an exposition and keeps the main ideas from getting lost amidst a morass of boring details and routine operations. By reminding the reader of small omissions in an unobtrusive fashion, such phrases help put the reader at ease --- if they are left out, it is easy for the reader to be thrown off-course by a missing step or be left with an uneasy feeling that there might be a hole in a proof. 54 | 55 | The dishonest use of these phrases occurs when a somewhat lengthy calculation has been left out because the author was too lazy to write it down. By using these phrases, the author hopes to intimidate potential critics from pointing out that material is missing by insinuating that anyone who would point out that something is missing is too stupid to fill in a few obvious steps. Frequent dishonest use of these terms may be a symptom of mathematheosis. 56 | 57 | The pedagogical use of these phrases occurs when the author has deliberately left the filling-in of missing steps as an exercise to the reader. 58 | %%%%% 59 | %%%%% 60 | \end{document} 61 | -------------------------------------------------------------------------------- /00A20-OpenQuestion.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{OpenQuestion} 4 | \pmcreated{2013-03-22 14:42:41} 5 | \pmmodified{2013-03-22 14:42:41} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{open question} 9 | \pmrecord{15}{36331} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmdefines{open problem} 16 | \pmdefines{open} 17 | \pmdefines{conjecture} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | \PMlinkescapeword{between} 44 | \PMlinkescapeword{force} 45 | \PMlinkescapeword{term} 46 | \PMlinkescapeword{terms} 47 | \PMlinkescapeword{difference} 48 | \PMlinkescapeword{even} 49 | 50 | The adjective \emph{open} is used by mathematicians to a statement which has neither been proven to be true or to be false. (In the light of incompleteness thorems, we should perhaps also add ``not been proven to be independent of the axioms'') 51 | 52 | Examples: 53 | 54 | ``It is an open question whether there are an infinite number of prime numbers all of whose digits are 1.'' means that it neither has been proven that there exist an infinity of such primes nor been proven that there are only a finite number of such primes. 55 | 56 | ``It is an open problem to determine the smallest number of lines which contain all points of the set $S$.'' means that the number in question has not yet been determined. 57 | 58 | The term \emph{conjecture} refers to a statement which the speaker has reason to believe the statement is correct even though the speaker cannot prove the statement. (Also, one should be warned about a common abuse of this term. Even after a statement has been proven, sometimes it is still referred to as a conjecture by force of habit.) 59 | 60 | The difference between the terms \emph{conjecture} and \emph{open question} is that the term \emph{open} is neutral --- saying that a statement is open does not connote that the speaker is voicing an opinion regarding the truth or falsity of the statement. 61 | 62 | The attachment below gives some examples of famous open problems in mathematics. \PMlinkexternal{Open Problem Garden}{http://garden.irmacs.sfu.ca/} is an external \PMlinkescapetext{link} that contains more examples of open problems in mathematics. 63 | %%%%% 64 | %%%%% 65 | \end{document} 66 | -------------------------------------------------------------------------------- /00A20-Pathological.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Pathological} 4 | \pmcreated{2013-03-22 14:41:56} 5 | \pmmodified{2013-03-22 14:41:56} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{pathological} 9 | \pmrecord{10}{36310} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{amsthm} 27 | 28 | \usepackage{mathrsfs} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | % 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | 43 | \newcommand{\sR}[0]{\mathbb{R}} 44 | \newcommand{\sC}[0]{\mathbb{C}} 45 | \newcommand{\sN}[0]{\mathbb{N}} 46 | \newcommand{\sZ}[0]{\mathbb{Z}} 47 | 48 | \usepackage{bbm} 49 | \newcommand{\Z}{\mathbbmss{Z}} 50 | \newcommand{\C}{\mathbbmss{C}} 51 | \newcommand{\R}{\mathbbmss{R}} 52 | \newcommand{\Q}{\mathbbmss{Q}} 53 | 54 | 55 | 56 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 57 | \newcommand*{\abs}[1]{| #1 |} 58 | 59 | 60 | 61 | \newtheorem{thm}{Theorem} 62 | \newtheorem{defn}{Definition} 63 | \newtheorem{prop}{Proposition} 64 | \newtheorem{lemma}{Lemma} 65 | \newtheorem{cor}{Corollary} 66 | \begin{document} 67 | In mathematics, a \emph{pathological object} is mathematical 68 | object that has a highly unexpected \PMlinkescapetext{property}. 69 | 70 | Pathological objects are typically percieved to, in some sense, be 71 | badly behaving. On the other hand, they are perfectly properly 72 | defined mathematical objects. Therefore this ``bad behaviour'' can 73 | simply be seen as a contradiction with our intuitive 74 | picture of how a certain object should behave. 75 | 76 | \subsubsection*{Examples} 77 | \begin{itemize} 78 | \item A very famous pathological function is the 79 | Weierstrass function, which is a continuous function 80 | that is nowhere differentiable. 81 | \item The Peano space filling curve. This pathological curve 82 | maps the unit interval $[0,1]$ continuously onto $[0,1]\times [0,1]$. 83 | \item The Cantor set. This is subset of the interval $[0,1]$ 84 | has the pathological property that it is uncountable 85 | yet its measure is zero. 86 | \item The Dirichlet's function from $\R$ to $\R$ is continuous at every 87 | irrational point and discontinuous at every rational point. 88 | \item Ackermann Function. 89 | \end{itemize} 90 | 91 | See also \cite{wiki}. 92 | 93 | \begin{thebibliography}{9} 94 | \bibitem{wiki}Wikipedia \PMlinkexternal{entry on pathological, mathematics}{http://en.wikipedia.org/wiki/Pathological (mathematics)}. 95 | \end{thebibliography} 96 | %%%%% 97 | %%%%% 98 | \end{document} 99 | -------------------------------------------------------------------------------- /00A20-Vacuous.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Vacuous} 4 | \pmcreated{2013-03-22 14:42:27} 5 | \pmmodified{2013-03-22 14:42:27} 6 | \pmowner{matte}{1858} 7 | \pmmodifier{matte}{1858} 8 | \pmtitle{vacuous} 9 | \pmrecord{9}{36326} 10 | \pmprivacy{1} 11 | \pmauthor{matte}{1858} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A20} 15 | \pmsynonym{vacuously}{Vacuous} 16 | \pmsynonym{vacuously true}{Vacuous} 17 | \pmsynonym{vacuous truth}{Vacuous} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | \usepackage{amsthm} 30 | 31 | \usepackage{mathrsfs} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | % 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | 46 | \newcommand{\sR}[0]{\mathbb{R}} 47 | \newcommand{\sC}[0]{\mathbb{C}} 48 | \newcommand{\sN}[0]{\mathbb{N}} 49 | \newcommand{\sZ}[0]{\mathbb{Z}} 50 | 51 | \usepackage{bbm} 52 | \newcommand{\Z}{\mathbbmss{Z}} 53 | \newcommand{\C}{\mathbbmss{C}} 54 | \newcommand{\R}{\mathbbmss{R}} 55 | \newcommand{\Q}{\mathbbmss{Q}} 56 | 57 | 58 | 59 | \newcommand*{\norm}[1]{\lVert #1 \rVert} 60 | \newcommand*{\abs}[1]{| #1 |} 61 | 62 | 63 | 64 | \newtheorem{thm}{Theorem} 65 | \newtheorem{defn}{Definition} 66 | \newtheorem{prop}{Proposition} 67 | \newtheorem{lemma}{Lemma} 68 | \newtheorem{cor}{Corollary} 69 | \begin{document} 70 | Suppose $X$ is a set and $P$ is a property defined as follows: 71 | \begin{eqnarray*} 72 | & & \mbox{$X$ has property $P$ if and only if} \\ 73 | & & \mbox{$\forall Y[$ $Y$ satisfies condition $1] \Rightarrow$ $Y$ satisfies condition $2$ } 74 | \end{eqnarray*} 75 | where condition $1$ and condition $2$ define the property. 76 | If condition $1$ is never satisfied then $X$ satisfies property $P$ 77 | \emph{vacuously}. 78 | 79 | \subsubsection*{Examples} 80 | \begin{enumerate} 81 | \item If $X$ is the set $\{1,2,3\}$ and $P$ is the property defined as above with condition $1=$ $Y$ is a infinite subset of $X$, and condition $2=$ $Y$ contains $7$. Then $X$ has property $P$ vacously; every infinite subset of $\{1,2,3\}$ contains the number $7$ \cite{wiki}. 82 | \item The empty set is a Hausdorff space (vacuously). 83 | \item Suppose property $P$ is defined by the statement\,: \\ 84 | \emph{The present King of France does not exist.}\\ 85 | Then either of the following propositions is satisfied vacuously. \\ 86 | \emph{The present king of France is bald.}\\ 87 | \emph{The present King of France is not bald.} 88 | \end{enumerate} 89 | 90 | \begin{thebibliography}{9} 91 | \bibitem{wiki}Wikipedia \PMlinkexternal{entry on Vacuous truth}{http://en.wikipedia.org/wiki/Vacuous_truth}. 92 | \end{thebibliography} 93 | %%%%% 94 | %%%%% 95 | \end{document} 96 | -------------------------------------------------------------------------------- /00A35-PreservationAndReflection.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PreservationAndReflection} 4 | \pmcreated{2013-03-22 17:12:18} 5 | \pmmodified{2013-03-22 17:12:18} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{preservation and reflection} 9 | \pmrecord{5}{39525} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A35} 15 | \pmdefines{preserve} 16 | \pmdefines{reflect} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb,amscd} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | \usepackage{mathrsfs} 24 | 25 | % used for TeXing text within eps files 26 | %\usepackage{psfrag} 27 | % need this for including graphics (\includegraphics) 28 | %\usepackage{graphicx} 29 | % for neatly defining theorems and propositions 30 | \usepackage{amsthm} 31 | % making logically defined graphics 32 | %%\usepackage{xypic} 33 | \usepackage{pst-plot} 34 | \usepackage{psfrag} 35 | 36 | % define commands here 37 | \newtheorem{prop}{Proposition} 38 | \newtheorem{thm}{Theorem} 39 | \newtheorem{ex}{Example} 40 | \newcommand{\real}{\mathbb{R}} 41 | \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} 42 | \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} 43 | \begin{document} 44 | In mathematics, the word ``preserve'' usually means the ``preservation of properties''. Loosely speaking, whenever a mathematical construct $A$ has some property $P$, after $A$ is somehow ``transformed'' into $A'$, the transformed object $A'$ also has property $P$. The constructs usually refer to sets and the transformations typically are functions or something similar. 45 | 46 | Here is a simple example, let $f$ be a function from a set $A$ to $B$. Let $A$ be a finite set. Let $P$ be the property of a set being finite. Then $f$ preserves $P$, since $f(A)$ is finite. Note that we are not saying that $B$ is finite. We are merely saying that the portion of $B$ that is the \emph{image} of $A$ (the transformed portion) is finite. 47 | 48 | Here is another example. The property of being connected in a topological space is preserved under a continuous function. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if $f:X\to Y$ is a continuous function from $X$ to $Y$. If $X$ is connected, so is $f(X)\subseteq Y$. 49 | 50 | Many more examples can be found in abstract algebra. Group homomorphisms, for example, preserve commutativity, as well as the property of being finitely generated. 51 | 52 | The word ``reflect'' is the dual notion of ``preserve''. It means that if the transformed object has property $P$, then the original object also has property $P$. This usage is rarely found outside of category theory, and is almost exclusively reserved for functors. For example, a faithful functor reflects isomorphism: if $F$ is a faithful functor from $\mathcal{C}$ to $\mathcal{D}$, and the object $F(A)$ is isomorphic to the object $F(B)$ in $\mathcal{D}$, then $A$ is isomorphic to $B$ in $\mathcal{C}$. 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /00A69-MathematicsOfFinance.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MathematicsOfFinance} 4 | \pmcreated{2013-03-22 16:40:17} 5 | \pmmodified{2013-03-22 16:40:17} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{mathematics of finance} 9 | \pmrecord{7}{38878} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Topic} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A69} 15 | \pmclassification{msc}{91B28} 16 | \pmclassification{msc}{00A06} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | 42 | \begin{document} 43 | From ancient times, a socially significant application of mathematics 44 | has been in accounting. This entry points to places on 45 | PlanetMath which deal with this topic. 46 | 47 | \begin{enumerate} 48 | \item interest 49 | \item simple interest 50 | \item compound interest, continuously compounded 51 | \item interest rate 52 | \item effective interest rate 53 | \item instantaneous effective interest rate 54 | \item present value, net present value 55 | \item coupon rate 56 | \item rate of return 57 | \item yield rate, yield curve 58 | \item force of interest 59 | \item annuity 60 | \item perpetuity 61 | \item call option, put option, 62 | \item American option, European option 63 | \item arbitrage, the arbitrage theorem 64 | \item risk neutral 65 | \item Black-Scholes formula 66 | \item capital asset pricing model (CAPM) 67 | \end{enumerate} 68 | 69 | %%%%% 70 | %%%%% 71 | \end{document} 72 | -------------------------------------------------------------------------------- /00A99-Arrow.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Arrow} 4 | \pmcreated{2013-03-22 12:55:46} 5 | \pmmodified{2013-03-22 12:55:46} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{arrow} 9 | \pmrecord{14}{33284} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | %\pmkeywords{arrow} 16 | %\pmkeywords{vector} 17 | %\pmkeywords{mapping} 18 | %\pmkeywords{function} 19 | %\pmkeywords{morphism} 20 | %\pmkeywords{functor} 21 | \pmrelated{Morphism} 22 | \pmrelated{Functor} 23 | \pmrelated{NaturalTransformation} 24 | \pmrelated{Mapping} 25 | \pmrelated{Vector} 26 | \pmrelated{Category} 27 | 28 | \endmetadata 29 | 30 | % this is the default PlanetMath preamble. as your knowledge 31 | % of TeX increases, you will probably want to edit this, but 32 | % it should be fine as is for beginners. 33 | 34 | % almost certainly you want these 35 | \usepackage{amssymb} 36 | \usepackage{amsmath} 37 | \usepackage{amsfonts} 38 | 39 | % used for TeXing text within eps files 40 | %\usepackage{psfrag} 41 | % need this for including graphics (\includegraphics) 42 | %\usepackage{graphicx} 43 | % for neatly defining theorems and propositions 44 | %\usepackage{amsthm} 45 | % making logically defined graphics 46 | %%%\usepackage{xypic} 47 | 48 | % there are many more packages, add them here as you need them 49 | 50 | % define commands here 51 | %\PMlinkescapeword{theory} 52 | \begin{document} 53 | An {\em arrow} is a line, usually straight, capped at one end, or both ends, with an arrowhead which points in a specific direction. In a flowchart, arrows show the order in which instructions are to be performed (including where branching may occur) or they show the flow of cause and effect. In certain arrangements of numbers, such as the Collatz tree, arrows show which numbers are the input values, with the output value being pointed at by the arrowhead. 54 | 55 | In writing about vectors, arrows can be written over the symbols for the origin and the endpoint, e.g., $\overrightarrow{AB}$ and $\overleftarrow{CD}$. \TeX{} provides commands for these. 56 | 57 | An arrow, regarded as a morphism, functor or natural transformation also plays central roles 58 | in Category Theory and Categorical Dynamics. 59 | %%%%% 60 | %%%%% 61 | \end{document} 62 | -------------------------------------------------------------------------------- /00A99-BlackboardBold.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{BlackboardBold} 4 | \pmcreated{2013-03-22 16:45:35} 5 | \pmmodified{2013-03-22 16:45:35} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{blackboard bold} 9 | \pmrecord{5}{38988} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | Certain capital letters are sometimes written in {\em blackboard bold} and considered to be a distinct symbol with a distinct meaning from their regular counterparts. This is to help alleviate overloading of the letters of the Latin-1 alphabet in mathematics and physics. For example, the blackboard letter $\mathbb{C}$ signifies the complex numbers, while in the same set of formulas one could also use $C$ as some variable or constant, complex or not, confident that it would not be confused with the complex numbers in general. 42 | 43 | Mathematicians are not unanimous in their approval of blackboard bold, however, and Donald Knuth did not include support for it in \TeX{}. The American Mathematical Society created the \TeX{} package \verb=amsfonts= which PlanetMath document preambles contain by default. The entire Latin-1 alphabet is available thus, while Unicode in Letterlike Symbols only makes available a subset of the most commonly used (such as $\mathbb{Q}$ and $\mathbb{Z}$). 44 | 45 | In typewritten dissertations, and even some published books, simple bold of non-italics letters is used. 46 | 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /00A99-EmptyProduct.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EmptyProduct} 4 | \pmcreated{2013-03-22 14:48:13} 5 | \pmmodified{2013-03-22 14:48:13} 6 | \pmowner{pahio}{2872} 7 | \pmmodifier{pahio}{2872} 8 | \pmtitle{empty product} 9 | \pmrecord{15}{36458} 10 | \pmprivacy{1} 11 | \pmauthor{pahio}{2872} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmrelated{EmptySet} 16 | \pmrelated{IndeterminateForm} 17 | \pmrelated{LocallyEuclidean} 18 | \pmrelated{AnalyticContinuationOfGammaFunction} 19 | \pmrelated{Introducing0thPower} 20 | \pmrelated{EmptySum} 21 | 22 | \endmetadata 23 | 24 | % this is the default PlanetMath preamble. as your knowledge 25 | % of TeX increases, you will probably want to edit this, but 26 | % it should be fine as is for beginners. 27 | 28 | % almost certainly you want these 29 | \usepackage{amssymb} 30 | \usepackage{amsmath} 31 | \usepackage{amsfonts} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | %\usepackage{amsthm} 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | \begin{document} 46 | The {\em empty product} of numbers is the borderline case of product, where the number of \PMlinkescapetext{factors is zero, i.e. the set of the factors} is empty. \,The most usual examples are the following. 47 | \begin{itemize} 48 | \item The \PMlinkname{zeroth power}{Introducing0thPower} of a non-zero number:\, $a^0$ 49 | \item The factorial of 0:\, 0! 50 | \item The \PMlinkname{prime factor presentation}{FundamentalTheoremOfArithmetics} of unity, which has no prime factors 51 | \end{itemize} 52 | The value of the empty sum of numbers is equal to the additive identity number, 0.\, Similarly, the empty product of numbers is equal to the 53 | \PMlinkname{multiplicative identity}{Unity} number, 1. 54 | 55 | \textbf{Note.}\, When considering the complex numbers as pairs of real numbers one often identifies the pairs $(x,\,0)$ and the reals $x$.\, In this sense one can think that the Cartesian product $\mathbb{R}\times\{0\}$ is equal to $\mathbb{R}$.\, This seems to \PMlinkescapetext{mean} the equation 56 | $$\mathbb{R}\times\mathbb{R}^0 = \mathbb{R}^{1+0} = \mathbb{R}^1 = \mathbb{R},$$ 57 | although the \PMlinkname{associativity}{GeneralAssociativity} of Cartesian product is nowhere stated.\, Nevertheless, it is sometimes natural to define that the Cartesian product of an empty collection of sets equals to a set with one element; so it may \PMlinkescapetext{mean} that e.g.\, $\mathbb{R}^0 = \{0\}.$ 58 | 59 | One can also consider empty products in categories. 60 | It follows directly from the definition that an object in a category 61 | is a \PMlinkname{product}{CategoricalDirectProduct} 62 | of an empty family of objects in the category 63 | if and only if it is a terminal object of the category. 64 | Sets are a special case of this: 65 | in the category of sets the singletons are the terminal objects, 66 | so the empty product exists and is a singleton. 67 | %%%%% 68 | %%%%% 69 | \end{document} 70 | -------------------------------------------------------------------------------- /00A99-JordansInequality.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{JordansInequality} 4 | \pmcreated{2013-03-22 11:42:29} 5 | \pmmodified{2013-03-22 11:42:29} 6 | \pmowner{Koro}{127} 7 | \pmmodifier{Koro}{127} 8 | \pmtitle{Jordan's inequality} 9 | \pmrecord{25}{30013} 10 | \pmprivacy{1} 11 | \pmauthor{Koro}{127} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmclassification{msc}{26D05} 16 | \pmrelated{ComparisonOfSinThetaAndThetaNearTheta0} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | %\usepackage{graphicx} 24 | %%%%%%%%%%%%%%%%%%%\usepackage{xypic} 25 | \begin{document} 26 | \PMlinkescapeword{states} 27 | 28 | \emph{Jordan's Inequality} states that 29 | $$ \frac{2}{\pi}x\leq\sin(x)\leq x $$ 30 | for all $x\in [0,\frac{\pi}{2}]$. 31 | %%%%% 32 | %%%%% 33 | %%%%% 34 | %%%%% 35 | %%%%% 36 | %%%%% 37 | %%%%% 38 | %%%%% 39 | %%%%% 40 | %%%%% 41 | %%%%% 42 | %%%%% 43 | %%%%% 44 | %%%%% 45 | %%%%% 46 | %%%%% 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /00A99-KochelNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{KochelNumber} 4 | \pmcreated{2013-03-22 16:06:58} 5 | \pmmodified{2013-03-22 16:06:58} 6 | \pmowner{CompositeFan}{12809} 7 | \pmmodifier{CompositeFan}{12809} 8 | \pmtitle{K\"ochel number} 9 | \pmrecord{5}{38182} 10 | \pmprivacy{1} 11 | \pmauthor{CompositeFan}{12809} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmsynonym{Kochel number}{KochelNumber} 16 | \pmrelated{OpusNumber} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | 42 | \begin{document} 43 | The {\em K\"ochel number} $K_n$ of a \PMlinkescapetext{composition} by Wolfgang Amadeus Mozart is its index in the ordered set compiled by Ludwig von K\"ochel in the conjectured chronological \PMlinkescapetext{order} of \PMlinkescapetext{composition}. ${{K_n} \over {25}} + 10$ approximates Mozart's age at time of writing reasonably well for $n > 100$ (thus ${{K_n} \over {25}} + 1766$ approximates the year). 44 | 45 | \begin{thebibliography}{5} 46 | \bibitem{pr} L. von K\"ochel, {\it Chronologisch-thematisches Verzeichnis s\"amtlicher Tonwerke Wolfgang Amade Mozarts; nebst Angabe der verlorengegangenen, angefangenen, \"ubertragenen, zweifelhaften und unterschobenen Kompositionen}, (1947), Ann Arbor: J. W. Edwards. 47 | \end{thebibliography} 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A99-OpusNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{OpusNumber} 4 | \pmcreated{2013-03-22 16:44:58} 5 | \pmmodified{2013-03-22 16:44:58} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{opus number} 9 | \pmrecord{4}{38974} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmrelated{KochelNumber} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | An {\em opus number} is a number attached to a particular musical composition or set of compositions by a publisher to denote the order of publication by the given composer. 43 | Theoretically, opus numbers would give musicologists a simple means to chart the growth of a composer chronologically, to make statements to the effect that there is a direct correlation between such and such composer's handling of a particular musical technique and the opus numbers. 44 | 45 | In practice, however, many factors contribute to making the opus numbers for certain composers less than useful, such as their being assigned by the composer rather than the publisher. Anton\'in Dvo\v{r}\'ak, for example, tried to fool publishers in an effort to get the best deal, and thus often gave misleadingly low or high opus numbers to his compositions. The confusion with Carl Nielsen's opus numbers arose not from any wheeling and dealing, but from his not having an exclusive publisher and not keeping track of which opus numbers he hadn't used (see sequence A113529 in Sloane's OEIS for a listing of opus numbers skipped over by Carl Nielsen). 46 | 47 | Other composers just never had any opus numbers given to their compositions by either publishers or themselves; musicologists then sometimes try to create a catalog with numbers named after themselves. In the case of Mozart, the K\"ochel numbers by Ludwig von K\"ochel are almost universally accepted, but in the case of Domenico Scarlatti, three different musicologists came up with three different catalogs, leading to many of Scarlatti's sonatas to be known by three different numbers. The lack of opus numbers is not a problem for composers whose repertory works consist of less than a dozen of large-scale works (e.g., Bruckner and Mahler). 48 | 49 | \begin{thebibliography}{1} 50 | \bibitem{hs} Sch\"onzeler, H. H. {\it Dvo\v{r}\'ak} London: M. Boyers (1984): 220 - 239 51 | \end{thebibliography} 52 | %%%%% 53 | %%%%% 54 | \end{document} 55 | -------------------------------------------------------------------------------- /00A99-QED.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{QED} 4 | \pmcreated{2013-03-22 12:40:14} 5 | \pmmodified{2013-03-22 12:40:14} 6 | \pmowner{mathwizard}{128} 7 | \pmmodifier{mathwizard}{128} 8 | \pmtitle{QED} 9 | \pmrecord{8}{32945} 10 | \pmprivacy{1} 11 | \pmauthor{mathwizard}{128} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmsynonym{Q.E.D}{QED} 16 | \pmrelated{QEDInTheoreticalAndMathematicalPhysics} 17 | \pmrelated{QCDOrQuantumChromodynamics} 18 | \pmrelated{MathematicalFoundationsOfQuantumFieldTheories} 19 | \pmrelated{QuantumOperatorAlgebrasInQuantumFieldTheories} 20 | \pmrelated{GrassmanHopfAlgebrasAndTheirDualCoAlgebras} 21 | \pmrelated{FoundationsOfQuantumFieldTheories} 22 | \pmrelated{QuantumChromod} 23 | \pmdefines{Halmos symbol} 24 | \pmdefines{tombstone} 25 | \pmdefines{Halmos tombstone} 26 | 27 | \endmetadata 28 | 29 | \usepackage{amssymb} 30 | \usepackage{amsmath} 31 | \usepackage{amsfonts} 32 | 33 | %\usepackage{psfrag} 34 | %\usepackage{graphicx} 35 | %%%\usepackage{xypic} 36 | \begin{document} 37 | The term ``QED'' is actually an abbreviation and stands for the Latin \emph{quod erat demonstrandum}, meaning ``which was to be demonstrated.'' 38 | 39 | QED typically is used to signify the end of a mathematical proof. The symbol 40 | 41 | $$ \square $$ 42 | 43 | is often used in place of ``QED,'' and is called the ``tombstone'', ``Halmos symbol'' or ``Halmos tombstone'' after mathematician Paul Halmos (it can vary in width, however, and sometimes it is fully or partially shaded). Halmos borrowed this symbol from magazines, where it was used to denote ``end of article''. 44 | %%%%% 45 | %%%%% 46 | \end{document} 47 | -------------------------------------------------------------------------------- /00A99-QHoldingBay47.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{QHoldingBay47} 4 | \pmcreated{2013-03-22 17:48:46} 5 | \pmmodified{2013-03-22 17:48:46} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Q holding bay 47} 9 | \pmrecord{7}{40276} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | The holding bay is now empty. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /00A99-SevensegmentDisplay.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SevensegmentDisplay} 4 | \pmcreated{2013-03-22 17:06:51} 5 | \pmmodified{2013-03-22 17:06:51} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{seven-segment display} 9 | \pmrecord{5}{39414} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | \usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | Most calculators use a {\em seven-segment display} to show both input and output numbers. Here is Champernowne's constant $C_{10}$ to 9 decimal places in a seven-segment display: 42 | 43 | \begin{center} 44 | \includegraphics{SevenSegmentDisplay} 45 | \end{center} 46 | 47 | Technically, the space for the decimal point counts as a segment, so for each place value there are actually eight segments. Generally, calculator manufacturers prefer the base 10 pandigital number 1234567980 as a demo number for the calculator packaging. 48 | 49 | Sometimes wear and tear can cause a segment to not light up or darken properly; this can occasionally lead to confusion between digits. The simplest way to test for this is to input $8 \times \frac{10^x - 1}{9}$, where $x$ is the the number of decimal place values available on the display (that is, a bunch of 8s). At various times in the past some manufacturers have developed variants of the glyphs for the digits to make it more obvious when a segment has burnt out, such as a 0 that looks like a lowercase O, or a 6 that looks like a lowercase B. Given that many scientific calculators offer the option of doing calculations in hexadecimal, the latter option is not acceptable for such calculators. The 7 on Sharp-brand calculators uses the upper left vertical segment. 50 | %%%%% 51 | %%%%% 52 | \end{document} 53 | -------------------------------------------------------------------------------- /00A99-SiegelsAxiom.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SiegelsAxiom} 4 | \pmcreated{2013-03-22 14:11:59} 5 | \pmmodified{2013-03-22 14:11:59} 6 | \pmowner{mathwizard}{128} 7 | \pmmodifier{mathwizard}{128} 8 | \pmtitle{Siegel's axiom} 9 | \pmrecord{8}{35631} 10 | \pmprivacy{1} 11 | \pmauthor{mathwizard}{128} 12 | \pmtype{Axiom} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | In a mathematical text two mathematical symbols that do not form a larger \PMlinkescapetext{complex} of \PMlinkescapetext{symbols} have to be \PMlinkescapetext{separated} by at least one word or by a punctuation mark.\\ 41 | \textbf{Example:} Do not write ``... take $n$ $n\times n$-matrices'', but rather ``take $n$ matrices of order $n\times n$''. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /00A99-Superscript.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Superscript} 4 | \pmcreated{2013-03-22 17:06:45} 5 | \pmmodified{2013-03-22 17:06:45} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{superscript} 9 | \pmrecord{7}{39412} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | A {\em superscript} is a symbol or group of symbols written above, and usually a bit to the right, of another symbol or group of symbols in order to show a relation among the two. The most common use of superscripts in mathematics is for notating exponentiation. For example, in the expression $4^7$, the 7 is a superscript to the 4. Iterated sum and product notation also uses superscripts most of the time, to show the iterator end values, and the same goes for integral notation. Notation of iterated functions also benefit from using superscripts to cut down on nested parentheses, e.g., $f^3(x) = f(f(f(x)))$; examples of iterated functions include the iterated totient function $\phi^i(n)$ and iterated sum of divisors function $\sigma^i(n)$. The contravariant indices of tensors are also denoted by superscripts. 42 | 43 | Nonfiction literature in general uses number superscripts to indicate footnotes with the corresponding numbers. This usage is often avoided in mathematical books and papers to prevent possible confusion with exponentiation. 44 | %%%%% 45 | %%%%% 46 | \end{document} 47 | -------------------------------------------------------------------------------- /00A99-TFAE.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TFAE} 4 | \pmcreated{2013-03-22 12:44:40} 5 | \pmmodified{2013-03-22 12:44:40} 6 | \pmowner{ariels}{338} 7 | \pmmodifier{ariels}{338} 8 | \pmtitle{TFAE} 9 | \pmrecord{4}{33048} 10 | \pmprivacy{1} 11 | \pmauthor{ariels}{338} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmsynonym{the following are equivalent}{TFAE} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} 42 | \newcommand{\norm}[1]{\left\|#1\right\|} 43 | 44 | % Some sets 45 | \newcommand{\Nats}{\mathbb{N}} 46 | \newcommand{\Ints}{\mathbb{Z}} 47 | \newcommand{\Reals}{\mathbb{R}} 48 | \newcommand{\Complex}{\mathbb{C}} 49 | \begin{document} 50 | The abbreviation ``TFAE'' is shorthand for ``\textbf{t}he \textbf{f}ollowing \textbf{a}re \textbf{e}quivalent''. It is used before a set of equivalent conditions (each implies all the others). 51 | 52 | In a definition, when one of the conditions is somehow ``better'' (simpler, shorter, ...), it makes sense to phrase the definition with that condition, and mention that the others are equivalent. ``TFAE'' is typically used when none of the conditions can take priority over the others. Actually proving the claimed equivalence must, of course, be done separately. 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /00A99-TextFigures.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TextFigures} 4 | \pmcreated{2013-03-22 17:06:48} 5 | \pmmodified{2013-03-22 17:06:48} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{text figures} 9 | \pmrecord{5}{39413} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | \usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | Some fonts have a {\em text figures} variant of the digits 0 to 9 where some of them have ascenders and some have descenders so that they blend better with the general text. Here is Champernowne's constant $C_{10}$ to 9 decimal places written in Georgia: 42 | 43 | \begin{center} 44 | \includegraphics{TextFiguresDemo} 45 | \end{center} 46 | 47 | Usually, 6 and 8 have ascenders, 3, 4, 5, 7 and 9 have descenders. Since most texts use base 10 numbers, and each digit usually occurs in about the same proportion as the other digits, this means that numbers have varying heights in just about the same propoertions as the words in the text. Using text figures for base 3 numbers would result in numbers always lacking in ascenders and descenders. For mathematical books and papers, however, text figures are generally eschewed. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /00A99-TopicEntryOnMiscellaneousMathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TopicEntryOnMiscellaneousMathematics} 4 | \pmcreated{2013-03-22 18:00:42} 5 | \pmmodified{2013-03-22 18:00:42} 6 | \pmowner{rm50}{10146} 7 | \pmmodifier{rm50}{10146} 8 | \pmtitle{topic entry on miscellaneous mathematics} 9 | \pmrecord{18}{40526} 10 | \pmprivacy{1} 11 | \pmauthor{rm50}{10146} 12 | \pmtype{Topic} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmclassification{msc}{00A20} 16 | %\pmkeywords{metamathematics} 17 | %\pmkeywords{beyond mathematics} 18 | %\pmkeywords{mathematical ontology} 19 | %\pmkeywords{mathematical logics} 20 | %\pmkeywords{categorical ontology} 21 | \pmrelated{Metamathematics} 22 | \pmrelated{AnOutlineOfHilbertsProgramme} 23 | \pmrelated{BibliographyForAxiomaticsAndMathematicsFoundationsInCategories} 24 | 25 | \endmetadata 26 | 27 | % this is the default PlanetMath preamble. as your knowledge 28 | % of TeX increases, you will probably want to edit this, but 29 | % it should be fine as is for beginners. 30 | 31 | % almost certainly you want these 32 | \usepackage{amssymb} 33 | \usepackage{amsmath} 34 | \usepackage{amsfonts} 35 | 36 | % used for TeXing text within eps files 37 | %\usepackage{psfrag} 38 | % need this for including graphics (\includegraphics) 39 | %\usepackage{graphicx} 40 | % for neatly defining theorems and propositions 41 | %\usepackage{amsthm} 42 | % making logically defined graphics 43 | %%%\usepackage{xypic} 44 | 45 | % there are many more packages, add them here as you need them 46 | 47 | % define commands here 48 | 49 | \begin{document} 50 | This topic provides links to those areas that are not restricted to Mathematics, but rather are metamathematical, or relevant to mathematics, though more general in nature. 51 | 52 | \begin{enumerate} 53 | \item Applied Mathematics;\\ 54 | some physical applications:\\ 55 | catacaustic, Heron's principle, uncertainty principle, motion in central-force field, heat equation, wave equation 56 | 57 | \item Definition(s) of Metamathematics 58 | \item \PMlinkname{Formal Logics and Metamathematics}{FormalLogicsAndMetaMathematics} 59 | \item Mathematical logics 60 | \item Theory of Models, AI, `Automatic' proof generation 61 | \item Problem solving techniques and guides 62 | \item Mathematics vocabulary 63 | \item Automatic translation and Meta-languages 64 | \item Categorical Ontology 65 | \item \PMlinkname{Does Mathematics need Philosophy?}{GWLeibnizSQuote} 66 | \item Mathematical philosophy and \PMlinkname{Philosophical mathematics}{BeyondFormalism} 67 | (Philosophy of Mathematics and Mathematics of Philosophy) 68 | \item Mathematical (Theoretical) physics and Physical mathematics 69 | \item Quotations about mathematics 70 | \item \PMlinkescapetext{Terms from foreign languages used in mathematics}: \PMlinkname{html version}{TermsFromForeignLanguagesUsedInMathematics}, \PMlinkname{page images version}{TermsFromForeignLanguagesUsedInMathematicsPageImagesVersion} 71 | \item Mathematics education 72 | \item Characterisation 73 | \item \PMlinkname{History of Mathematics}{PlatosMathematics} and \PMlinkname{mathematical concepts}{AnOutlineOfHilbertsProgramme} 74 | \item Biographies on PlanetMath 75 | \item Resources for Mathematics: bibliographies, books, textbooks, mathematics resources on the web, computer softare/programs, web links to other encyclopedic references concerned with, or related to, Mathematics 76 | \item Mathematical applications, paradoxes and puzzles 77 | \end{enumerate} 78 | %%%%% 79 | %%%%% 80 | \end{document} 81 | -------------------------------------------------------------------------------- /00A99-TriceSequence.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TriceSequence} 4 | \pmcreated{2013-03-22 16:45:00} 5 | \pmmodified{2013-03-22 16:45:00} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{Trice sequence} 9 | \pmrecord{5}{38975} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | The {\em Trice sequence} consists of the nine numbers 21, 36, 55, 60, 67, 68, 92, 93, 125 in ascending order. It was published in Clifford Pickover's {\it Mazes of the Mind} in 1992 and added to the On-Line Encyclopedia of Integer Sequences early on in its history (see A001491), but for years people couldn't figure out what the sequence was because most of them were searching for a mathematical explanation. The first three numbers are both triangular and hexagonal, but none of the following numbers are. According to Neil Sloane, ``this was a mystery for many years; even Clifford Pickover could not recall the explanation .'' It wasn't until 1998 that Derek Holt identified the sequence and let Sloane know that it consists of the opus numbers of Beethoven's nine symphonies. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /00A99-Triskaidekaphobia.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Triskaidekaphobia} 4 | \pmcreated{2013-03-22 17:23:48} 5 | \pmmodified{2013-03-22 17:23:48} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{triskaidekaphobia} 9 | \pmrecord{7}{39765} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | \PMlinkescapeword{irrational} 42 | \PMlinkescapeword{join} 43 | \PMlinkescapeword{flag} 44 | \PMlinkescapeword{stars} 45 | \PMlinkescapeword{floor} 46 | 47 | \emph{Triskaidekaphobia} is the irrational fear of the number 13. This fear is recognized as a mental disorder by the Diagnostic and Statistical Manual of Mental Disorders IV (DSM-IV). Few mathematicians suffer from triskaidekaphobia; generally it's artists who are afflicted. For example, the composer Arnold Schoenberg and the pianist Glenn Gould. However, Benjamin Franklin was mildly afflicted: if there were 13 people at his dinner table, he would call for his secretary to join the party. He was apparently not bothered by the fact that the United States at the time consisted of 13 states and the flag had 13 stars and 13 stripes. The rationale for Franklin's affliction seems to have been that there were 13 people at the Last Supper of Jesus Christ. Architects for the most part don't have to worry about triskaidekaphobia when building new buildings, but when working on existing buildings they have to remember to subtract one if the building's 13th floor was renumbered 14. 48 | 49 | \begin{thebibliography}{1} 50 | \bibitem{wc} W. Clarke, \PMlinkexternal{{\it Some Mathematical History: The History of Numbers}}{http://faculty.lasierra.edu/~wclarke/hist.pdf} (2003): 7 51 | \end{thebibliography} 52 | %%%%% 53 | %%%%% 54 | \end{document} 55 | -------------------------------------------------------------------------------- /00A99-WLOG.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{WLOG} 4 | \pmcreated{2013-03-22 12:40:17} 5 | \pmmodified{2013-03-22 12:40:17} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{WLOG} 9 | \pmrecord{6}{32946} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{00A99} 15 | \pmsynonym{WOLOG}{WLOG} 16 | \pmsynonym{without loss of generality}{WLOG} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | 24 | %\usepackage{psfrag} 25 | %\usepackage{graphicx} 26 | %%%\usepackage{xypic} 27 | \begin{document} 28 | ``WLOG'' (or ``WOLOG'') is an acronym which stands for ``without loss of generality.'' 29 | 30 | WLOG is invoked in situations where some property of a model or system is invariant under the particular choice of instance attributes, but for the sake of demonstration, these attributes must be fixed. 31 | 32 | For example, we might want to prove something about open intervals $(a, b)$ of the real number line. But the proof might become too tedious if $a$ and $b$ were arbitrary real numbers, so in the proof we simply assume that $a = 0$ and $b = 1$, and \emph{without loss of generality}, the same arguments apply to general intervals $(a, b)$. Depending on the proof, the loss of generality might be accomplished by translating and scaling the interval to $(0,1)$ \emph{before} carrying out the argument, and then translating and rescaling back to $(a, b)$ 33 | afterwards. 34 | 35 | WLOG can also be invoked to shorten proofs where there are a number of choices of configuration, but the proof is ``the same'' for each of them. We need only walk through the proof for one of these configurations, and ``WLOG'' serves as a note that we haven't weakened the argument. For example, the proof of the fundamental theorem of arithmetic uses this notion, in essence settling on a ``canonical form'' for prime factorizations to simplify the argument. 36 | 37 | For more examples, see \PMlinkexternal{approximate index of PM entries invoking WLOG}{http://planetmath.org/?op=search\&term=WLOG+without+loss+generality}. 38 | %%%%% 39 | %%%%% 40 | \end{document} 41 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # 00_General 2 | 3 | [![Join the chat at https://gitter.im/planetmath/00_General](https://badges.gitter.im/planetmath/00_General.svg)](https://gitter.im/planetmath/00_General?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge) 4 | 5 | - 00-XX: General 6 | - 00-01: Instructional exposition (textbooks, tutorial papers, etc.) 7 | - 00-02: Research exposition (monographs, survey articles) 8 | - 00Axx: General and miscellaneous specific topics 9 | - 00A05: General mathematics 10 | - 00A06: Mathematics for nonmathematicians (engineering, social sciences, etc.) 11 | - 00A07: Problem books 12 | - 00A08: Recreational mathematics [See also 97A20] 13 | - 00A09: Popularization of mathematics 14 | - 00A15: Bibliographies 15 | - 00A17: External book reviews 16 | - 00A20: Dictionaries and other general reference works 17 | - 00A22: Formularies 18 | - 00A30: Philosophy of mathematics [See also 03A05] 19 | - 00A35: Methodology of mathematics, didactics [See also 97Cxx, 97Dxx] 20 | - 00A65: Mathematics and music 21 | - 00A66: Mathematics and visual arts, visualization 22 | - 00A67: Mathematics and architecture 23 | - 00A69: General applied mathematics {For physics, see 00A79 and Sections 70 through 86} 24 | - 00A71: Theory of mathematical modeling 25 | - 00A72: General methods of simulation 26 | - 00A73: Dimensional analysis 27 | - 00A79: Physics (use more specific entries from Sections 70 through 86 when possible) 28 | - 00A99: Miscellaneous topics 29 | - 00Bxx: Conference proceedings and collections of papers 30 | - 00B05: Collections of abstracts of lectures 31 | - 00B10: Collections of articles of general interest 32 | - 00B15: Collections of articles of miscellaneous specific content 33 | - 00B20: Proceedings of conferences of general interest 34 | - 00B25: Proceedings of conferences of miscellaneous specific interest 35 | - 00B30: Festschriften 36 | - 00B50: Volumes of selected translations 37 | - 00B55: Miscellaneous volumes of translations 38 | - 00B60: Collections of reprinted articles [See also 01A75] 39 | - 00B99: None of the above, but in this section 40 | -------------------------------------------------------------------------------- /pdf/00-02-GeorgePolya.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/planetmath/00_General/f6b6fd1a507bf68001255dd23e173d82e41fb9ff/pdf/00-02-GeorgePolya.pdf 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