This second edition of "Categories Work" adds two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them--items of interest in their own right and also in view of their use in string theory in quantum field theory. The second new chapter describes 2-categories and the higher-dimensional categories that have recently come into prominence. In addition, the bibliography has been expanded to cover some of the many other recent advances concerning categories.
此书为英文版。
Preface
Notationandconventions
Ⅰ AFirstCourseinNumberTheory
1 DivisibilityandPrimes
1.1 DivisionAlgorithm
1.2 GreatestCommonDivisors
1.3 TheEuclideanAlgorithmandContinuedFractions
1.4 TheFundamentalTheoremofArithmetic
1.5 Euclid'sTheoremandtheSieveofEratosthenes
1.6 ALinearDiophantineEquation
1.7 Notes
2 Congruences
2.1 TheRingofCongruenceClasses
2.2 LinearCongruences
2.3 TheEulerPhiFunction
2.4 ChineseRemainderTheorem
2.5 Euler'sTheoremandFermat'sTheorem
2.6 PseudoprimesandCarmichaelNumbers
2.7 PublicKeyCryptography
2.8 Notes
3 PrimitiveRootsandQuadraticReciprocity
3.1 PolynomialsandPrimitiveRoots
3.2 PrimitiveRootstoCompositeModuli
3.3 PowerResidues
3.4 QuadraticResidues
3.5 QuadraticReciprocityLaw
3.6 QuadraticResiduestoCompositeModuli
3.7 Notes
4 FourierAnalysisonFiniteAbelianGroups
4.1 TheStructureofFiniteAbelianGroups
4.2 CharactersofFiniteAbelianGroups
4.3 ElementaryFourierAnalysis
4.4 PoissonSummation
4.5 TraceFormulaeonFiniteAbelianGroups
4.6 GaussSumsandQuadraticReciprocity
4.7 TheSignoftheGaussSum
4.8 Notes
5 TheabcConjecture
5.1 IdealsandRadicals
5.2 Derivations
5.3 Mason'sTheorem
5.4 TheabcConjecture
5.5 TheCongruenceabcConjecture
5.6 Notes
Ⅱ DivisorsandPrimesinMultiplicativeNumberTheory
6 ArithmeticFunctions
6.1 TheRingofArithmeticFunctions
6.2 MeanValuesofArithmeticFunctions
6.3 TheMSbiusFunction
6.4 MultiplicativeFunctions
6.5 ThemeanvalueoftheEulerPhiFunction
6.6 Notes
7 DivisorFunctions
7.1 DivisorsandFactorizations
7.2 ATheoremofRamanujan
7.3 SumsofDivisors
7.4 SumsandDifferencesofProducts
7.5 SetsofMultiples
7.6 AbundantNumbers
7.7 Notes
8 PrimeNumbers
8.1 Chebyshev'sTheorems
8.2 Mertens'sTheorems
8.3 TheNumberofPrimeDivisorsofanInteger
8.4 Notes
9 ThePrimeNumberTheorem
9.1 GeneralizedVonMangoldtFunctions
9.2 Selberg'sFormulae
9.3 TheElementaryProof
9.4 IntegerswithkPrimeFactors
9.5 Notes
10 PrimesinArithmeticProgressions
10.1 DirichletCharacters
10.2 DirichletL-Functions
10.3 PrimesModulo4
10.4 TheNonvanishingofL(1,X)
10.5 Notes
Ⅲ ThreeProblemsinAdditiveNumberTheory
11 Waring'sProblem
11.1 SumsofPowers
11.2 StableBases
11.3 Shnirel'man'sTheorem
11.4 Waring'sProblemforPolynomials
11.5 Notes
12 SumsofSequencesofPolynomials
12.1 SumsandDifferencesofWeightedSets
12.2 LinearandQuadraticEquations
12.3 AnUpperBoundforRepresentations
12.4 Waring'sProblemforSequencesofPolynomials
12.5 Notes
13 Liouville'sIdentity
13.1 AMiraculousFormula
13.2 PrimeNumbersandQuadraticForms
13.3 ATernaryForm
13.4 ProofofLiouville'sIdentity
13.5 TwoCorollaries
13.6 Notes
14 SumsofanEvenNumberofSquares
14.1 SummaryofResults
14.2 ARecursionFormula
14.3 SumsofTwoSquares
14.4 SumsofFourSquares
14.5 SumsofSixSquares
14.6 SumsofEightSquares
14.7 SumsofTenSquares
14.8 Notes
15 PartitionAsymptotics
15.1 TheSizeofp(n)
15.2 PartitionFunctionsforFiniteSets
15.3 UpperandLowerBoundsforlogp(n)
15.4 Notes
16 AnInverseTheoremforPartitions
16.1 DensityDeterminesAsymptotics
16.2 AsymptoticsDetermineDensity
16.3 AbelianandTauberianTheorems
16.4 Notes
References
Index
