Generating functionsFrom Wikipedia, the free encyclopediaContents1 Almost convergent sequence 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Arithmetic progression 22.1 Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Betti number 63.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Example 1: Betti numbers of a simplicial complex K. . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Example 2: the rst Betti number in graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Relationship with dimensions of spaces of dierential forms . . . . . . . . . . . . . . . . . . . . . 93.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Cauchy product 114.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.1 Cauchy product of two nite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.2 Cauchy product of two innite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.3 Cauchy product of two nite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.4 Cauchy product of two innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.5 Cauchy product of two power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Convergence and Mertens theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12iii CONTENTS4.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3.2 Proof of Mertens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.2 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 Cesros theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6.1 Products of nitely many innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Relation to convolution of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Cauchy sequence 185.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.1 In topological vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4.5 In a hyperreal continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Chebyshevs sum inequality 236.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Continuous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Complementary sequences 257.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Properties of complementary pairs of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 Golay pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27CONTENTS iii7.5 Applications of complementary sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Cumulant-generating function 298.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.1.1 Alternative denition of the cumulant generating function . . . . . . . . . . . . . . . . . . 298.2 Uses in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3 Cumulants of some discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . 308.4 Cumulants of some continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . 318.5 Some properties of the cumulant generating function . . . . . . . . . . . . . . . . . . . . . . . . . 318.6 Some properties of cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6.1 Invariance and equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6.2 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6.3 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6.4 A negative result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.6.5 Cumulants and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.6.6 Relation to moment-generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.6.7 Cumulants and set-partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.6.8 Cumulants and combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.7 Joint cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.7.1 Conditional cumulants and the law of total cumulance . . . . . . . . . . . . . . . . . . . . 378.8 Relation to statistical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.10 Cumulants in generalized settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.10.1 Formal cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.10.2 Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.10.3 Cumulants of a polynomial sequence of binomial type . . . . . . . . . . . . . . . . . . . . 398.10.4 Free cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Cutting sequence 419.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110Cyclic sieving 4210.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311DavenportSchinzel sequence 4411.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44iv CONTENTS11.2Length bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.3Application to lower envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712Disjunctive sequence 4812.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2Rich numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013Divisibility sequence 5113.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114Ducci sequence 5314.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.3Modulo two form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.4Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.5Other related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515Examples of generating functions 5615.1Worked example A: basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.1.1 Bivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.2Worked example B: Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716Factorial moment generating function 5816.1Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917Farey sequence 6017.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.3.1 Sequence length and index of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.3.2 Farey neighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6417.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.3.4 Ford circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65CONTENTS v17.3.5 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.4Next term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6717.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818Generating function 6918.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6918.1.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6918.1.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.1.3 Poisson generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.1.4 Lambert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.1.5 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7018.1.6 Dirichlet series generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.1.7 Polynomial sequence generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.2Ordinary generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7118.2.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7218.2.2 Multiplication yields convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.2.3 Relation to discrete-time Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 7318.2.4 Asymptotic growth of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7318.2.5 Bivariate and multivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . 7418.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.3.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.3 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.4 Dirichlet series generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.5 Multivariate generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.4.1 Techniques of evaluating sums with generating function . . . . . . . . . . . . . . . . . . . 7618.4.2 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7618.4.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.5Other generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919Geometric progression 8019.1Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.2Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82vi CONTENTS19.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.2.3 Innite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8319.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8519.3Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8519.4Relationship to geometry and Euclids work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8719.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8720Halton sequence 8820.1Example of Halton sequence used to generate points in (0, 1) (0, 1) in R2. . . . . . . . . . . . . 8820.2Implementation in Pseudo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9021Harmonic progression (mathematics) 9121.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9121.2Use in geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9121.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9121.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222Innite product 9322.1Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.2Product representations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9422.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9422.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9522.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9523Interleave sequence 9623.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9623.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624Iterated function 9724.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9724.2Abelian property and Iteration sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9724.3Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.4Limiting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.5Fractional iterates and ows, and negative iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.6Some formulas for fractional iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100CONTENTS vii24.7Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10024.8Markov chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10024.9Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.10Means of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.11In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.12Denitions in terms of iterated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.13Lies data transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10124.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10224.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225Katydid sequence 10325.1Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326Limit of a sequence 10426.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.2Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.2.2 Formal Denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10626.3Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.4Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.5Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.6Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10726.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10826.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10826.8.1 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10826.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10926.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10927List of sums of reciprocals 11027.1Finitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11027.2Innitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11127.2.1 Convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11127.2.2 Divergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11227.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112viii CONTENTS27.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11228Logarithmically concave sequence 11328.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11328.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329Low-discrepancy sequence 11429.1Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.1.1 Low-discrepancy sequences in numerical integration . . . . . . . . . . . . . . . . . . . . 11429.2Denition of discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.3The KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.4The formula of Hlawka-Zaremba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.5The L2version of the KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.6The ErdsTurnKoksma inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.7The main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11729.8Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.9Construction of low-discrepancy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.9.1 Random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11829.9.2 Additive recurrence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11929.9.3 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11929.9.4 van der Corput sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.9.5 Halton sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.9.6 Hammersley set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12029.9.7 Poisson disk sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.10Graphical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12129.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130Mathematics of oscillation 12930.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.1.1 Oscillation of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12930.1.2 Oscillation of a function on an open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.1.3 Oscillation of a function at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.3Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13030.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13230.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13231Matsushimas formula 13331.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13332Moment-generating function 134CONTENTS ix32.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13432.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.3Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.3.1 Sum of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.3.2 Vector-valued random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13532.4Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.4.1 Calculations of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.5Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.6Relation to other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13632.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13732.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13733Monotone convergence theorem 13833.1Convergence of a monotone sequence of real numbers . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.3 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.1.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13833.2Convergence of a monotone series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.3Lebesgues monotone convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.3.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14033.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14233.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14234Periodic sequence 14334.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14334.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14334.3Periodic 0, 1 sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14434.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14435Polynomial sequence 14535.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14535.2Classes of polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14635.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14635.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636Polyphase sequence 14736.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14736.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147x CONTENTS37Probability-generating function 14837.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.1.1 Univariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.1.2 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.2.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14837.2.2 Probabilities and expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14937.2.3 Functions of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 14937.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.4Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15137.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15137.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138Random sequence 15238.1Early history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.2Modern approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15438.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15438.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439Rook polynomial 15539.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15539.1.1 Complete boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.2Matching polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.3Connection to matrix permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.4Complete rectangular boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.4.1 Rooks problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639.4.2 The rook polynomial as a generalization of the rooks problem. . . . . . . . . . . . . . . . 15739.4.3 Symmetric arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839.4.4 Arrangements counted by symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . 15939.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15940Sequence 16140.1Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16240.1.1 Important examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16240.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.2Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16440.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16540.2.3 Increasing and decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165CONTENTS xi40.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16540.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16540.3Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16640.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16740.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16740.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16840.4Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16840.5Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16940.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16940.5.2 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16940.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17040.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17040.5.5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17140.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17140.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17140.6Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17140.7Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17240.8Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17240.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17240.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17240.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17341Sequence space 17441.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17441.1.1 pspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17441.1.2 c and c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17541.1.3 Other sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17541.2Properties of pspaces and the space c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17641.2.1 pspaces are increasing in p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17741.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17741.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17742Shift rule 17842.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17843Sobol sequence 17943.1Good distributions in the s-dimensional unit hypercube . . . . . . . . . . . . . . . . . . . . . . . 17943.2A fast algorithm for the construction of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . 18043.3Additional uniformity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18043.4The initialization of Sobol numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18143.5Implementation and availability of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . . . 18143.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181xii CONTENTS43.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18243.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18243.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18244Stationary sequence 18344.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18344.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345Sturmian word 18445.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.1.1 Combinatoric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.1.2 Geometric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.2.2 Balanced aperiodic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.2.3 Slope and intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.2.4 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.3Non-binary words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.4Associated real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.5History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18845.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846Subadditivity 18946.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18946.3Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19046.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147Subsequence 19247.1Common subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.3Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19348Subsequential limit 19449Superadditivity 19549.1Examples of superadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195CONTENTS xiii49.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19650Tuple 19750.1Etymology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.1.1 Names for tuples of specic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.3Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.3.1 Tuples as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.3.2 Tuples as nested ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.3.3 Tuples as nested sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19950.4 n-tuples of m-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19950.5Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19950.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20050.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20050.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20151Van der Corput sequence 20251.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20351.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20351.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20351.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20352Vites formula 20452.1Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20552.2Interpretation and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20552.3Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20652.4Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20652.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20752.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20853Weisners method 20953.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20953.2Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 21053.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21053.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21453.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Chapter 1Almost convergent sequenceA bounded real sequence (xn) is said to be almost convergent to L if each Banach limit assigns the same value L tothe sequence (xn) .Lorentz proved that (xn) is almost convergent if and only iflimpxn +. . . +xn+p1p= Luniformly in n .The above limit can be rewritten in detail as( > 0)(p0)(p > p0)(n)xn +. . . +xn+p1pL < .Almost convergence is studied in summability theory. It is an example of a summability method which cannot berepresented as a matrix method.1.1 ReferencesG. Bennett and N.J. Kalton: Consistency theorems for almost convergence. Trans. Amer. Math. Soc.,198:23-43, 1974.J. Boos: Classical and modern methods in summability. Oxford University Press, New York, 2000.J. Connor and K.-G. Grosse-Erdmann: Sequential denitions of continuity for real functions. Rocky Mt. J.Math., 33(1):93-121, 2003.G.G. Lorentz: A contribution to the theory of divergent sequences. Acta Math., 80:167-190, 1948.This article incorporates material fromAlmost convergent on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.1Chapter 2Arithmetic progressionIn mathematics, an arithmetic progression(AP) or arithmetic sequence is a sequence of numbers such that thedierence between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 is an arithmeticprogression with common dierence of 2.If the initial term of an arithmetic progression is a1 and the common dierence of successive members is d, then thenth term of the sequence ( an ) is given by:an= a1 + (n 1)d,and in generalan= am + (n m)d.A nite portion of an arithmetic progression is called a nite arithmetic progression and sometimes just called anarithmetic progression. The sum of a nite arithmetic progression is called an arithmetic series.The behavior of the arithmetic progression depends on the common dierence d. If the common dierence is:Positive, the members (terms) will grow towards positive innity.Negative, the members (terms) will grow towards negative innity.2.1 SumThis section is about Finite arithmetic series. For Innite arithmetic series, see Innite arithmetic series.Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, theresulting sequence has a single repeated value in it, equal to the sum of the rst and last numbers (2 + 14 = 16). Thus16 5 = 80 is twice the sum.The sum of the members of a nite arithmetic progression is called an arithmetic series. For example, consider thesum:2 + 5 + 8 + 11 + 14This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of therst and last number in the progression (here 2 + 14 = 16), and dividing by 2:n(a1 +an)222.2. PRODUCT 3In the case above, this gives the equation:2 + 5 + 8 + 11 + 14 =5(2 + 14)2=5 162= 40.This formula works for any real numbers a1 and an . For example:_32_+_12_+12=3_32+12_2= 32.2.1.1 DerivationTo derive the above formula, begin by expressing the arithmetic series in two dierent ways:Sn= a1 + (a1 +d) + (a1 + 2d) + + (a1 + (n 2)d) + (a1 + (n 1)d)Sn= (an (n 1)d) + (an (n 2)d) + + (an 2d) + (an d) +an.Adding both sides of the two equations, all terms involving d cancel:2Sn= n(a1 +an).Dividing both sides by 2 produces a common form of the equation:Sn=n2(a1 +an).An alternate form results from re-inserting the substitution:an= a1 + (n 1)d :Sn=n2[2a1 + (n 1)d].Furthermore the mean value of the series can be calculated via:Sn/n :n =a1 +an2.In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics andIndian astronomy, gave this method in the Aryabhatiya (section 2.18).2.2 ProductThe product of the members of a nite arithmetic progression with an initial element a1, common dierences d, andn elements in total is determined in a closed expressiona1a2 an= da1dd(a1d+ 1)d(a1d+ 2) d(a1d+n 1) = dn_a1d_n= dn(a1/d +n)(a1/d),where xndenotes the rising factorial and denotes the Gamma function. (Note however that the formula is not validwhen a1/d is a negative integer or zero.)This is a generalization from the fact that the product of the progression 1 2 n is given by the factorial n!and that the product4 CHAPTER 2. ARITHMETIC PROGRESSIONm(m+ 1) (m+ 2) (n 2) (n 1) nfor positive integers m and n is given byn!(m1)!.Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n1)(5)up to the 50th term isP50= 550(3/5 + 50)(3/5) 3.78438 1098.2.3 Standard deviationThe standard deviation of any arithmetic progression can be calculated via:= |d|(n 1)(n + 1)12where n is the number of terms in the progression, and d is the common dierence between terms2.4 IntersectionsThe intersection of any two doubly-innite arithmetic progressions is either empty or another arithmetic progression,which can be found using the Chinese remainder theorem. If each two progressions in a family of doubly-innitearithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is,innite arithmetic progressions form a Helly family.[1] However, the intersection of innitely many innite arithmeticprogressions might be a single number rather than itself being an innite progression.2.5 Formulas at a GlanceIfa1andnSnnthenan= a1 + (n 1)d,an= am + (n m)d.Sn=n2[2a1 + (n 1)d].Sn=n(a1 +an)22.6. SEE ALSO 55.n = Sn/nn =a1 +an2.2.6 See alsoArithmetico-geometric sequenceGeneralized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowingseveral possible dierences.Harmonic progressionHeronian triangles with sides in arithmetic progressionProblems involving arithmetic progressionsUtonality2.7 References[1] Duchet, Pierre (1995), Hypergraphs, in Graham, R. L.; Grtschel, M.; Lovsz, L., Handbook of combinatorics, Vol. 1,2, Amsterdam: Elsevier, pp. 381432, MR 1373663. See in particular Section 2.5, Helly Property, pp. 393394.Sigler, Laurence E. (trans.) (2002). Fibonaccis Liber Abaci. Springer-Verlag. pp. 259260. ISBN 0-387-95419-8.2.8 External linksHazewinkel, Michiel, ed. (2001), Arithmetic series, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Arithmetic progression, MathWorld.Weisstein, Eric W., Arithmetic series, MathWorld.Chapter 3Betti numberIn algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity ofn-dimensional simplicial complexes. For the most reasonable nite-dimensional spaces (such as compact manifolds,nite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onward (Bettinumbers vanish above the dimension of a space), and they are all nite.A torus has one connected component (b0), two circular holes (b1,the one in the center and the one in the middle of the donut),and one two-dimensional void (b2, the inside of the donut) yielding Betti numbers of 1 (b0),2 (b1),1 (b2).The nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximumamountof cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc.[1] These numbers areused today in elds such as simplicial homology, computer science, digital images, etc.The term Betti numbers was coined by Henri Poincar after Enrico Betti.3.1 DenitionInformally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. The rstfew Betti numbers have the following denitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial63.2. EXAMPLE 1: BETTI NUMBERS OF A SIMPLICIAL COMPLEX K 7complexes:b0 is the number of connected componentsb1 is the number of one-dimensional or circular holesb2 is the number of two-dimensional voids or cavitiesThe two-dimensional Betti numbers are easier to understand because we see the world in 0, 1, 2, and 3-dimensions,however. The following Betti numbers are higher-dimensional than apparent physical space.For a non-negative integer k, the kth Betti number bk(X) of the space X is dened as the rank (number of linearlyindependent generators) of the abelian group Hk(X), the kth homology group of X. The kth homology group isHk= ker k/Imk+1 , the ks are the boundary maps of the simplicial complex and the rank of H is the kth Bettinumber. Equivalently, one can dene it as the vector space dimension of Hk(X; Q) since the homology group in thiscase is a vector space over Q. The universal coecient theorem, in a very simple torsion-free case, shows that thesedenitions are the same.More generally, given a eld F one can dene bk(X, F), the kth Betti number with coecients in F, as the vectorspace dimension of Hk(X, F).3.2 Example 1: Betti numbers of a simplicial complex KLet us go through a simple example of how to compute the Betti numbers for a simplicial complex.Here we have a simplicial complex with 0-simplices: a,b,c, and d, 1-simplices: E,F,G,H and I, and the only 2-simplexis J, which is the shaded region in the gure.It is clear that there is one connected component in this gure (b0),one hole, which is the shaded region (b1) and no voids or cavities (b2).This means that the rank of H0 is 1, the rank of H1 is 1 and the rank of H2 is 0.The Betti number sequence for this gure is 1,1,0,0,...; the Poincar polynomial is 1 +x3.3 Example 2: the rst Betti number in graph theoryIn topological graph theory the rst Betti number of a graph G with n vertices, m edges and k connected componentsequals8 CHAPTER 3. BETTI NUMBERmn +k.This may be proved straightforwardly by mathematical induction on the number of edges. A new edge either incre-ments the number of 1-cycles or decrements the number of connected components.The rst Betti number is also called the cyclomatic numbera term introduced by Gustav Kirchho before Bettispaper.[2] See cyclomatic complexity for an application to software engineering.The zero-th Betti number of a graph is simply the number of connected components k.[3]3.4 PropertiesThe (rational) Betti numbers bk(X) do not take into account any torsion in the homology groups, but they are veryuseful basic topological invariants. In the most intuitive terms, they allowone to count the number of holes of dierentdimensions.For a nite CW-complex K we have(K) =i=0(1)ibi(K, F),where (K) denotes Euler characteristic of K and any eld F.For any two spaces X and Y we havePXY= PXPY,where PX denotes thePoincarpolynomial of X, (more generally, the Poincar series, for innite-dimensionalspaces), i.e. the generating function of the Betti numbers of X:PX(z) = b0(X) +b1(X)z +b2(X)z2+ ,see Knneth theorem.If X is n-dimensional manifold, there is symmetry interchanging k and n k, for any k:bk(X) = bnk(X),under conditions (a closed and oriented manifold); see Poincar duality.The dependence on the eld F is only through its characteristic. If the homology groups are torsion-free, the Bettinumbers are independent of F. The connection of p-torsion and the Betti number for characteristic p, for p a primenumber, is given in detail by the universal coecient theorem (based on Tor functors, but in a simple case).3.5 Examples1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;the Poincar polynomial is1 +x2. The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... .3.6. RELATIONSHIP WITH DIMENSIONS OF SPACES OF DIFFERENTIAL FORMS 9the Poincar polynomial is(1 +x)3= 1 + 3x + 3x2+x33. Similarly, for an n-torus,the Poincar polynomial is(1 +x)n(by the Knneth theorem), so the Betti numbers are the binomial coecients.It is possible for spaces that are innite-dimensional in an essential way to have an innite sequence of non-zero Bettinumbers. An example is the innite-dimensional complex projective space, with sequence 1, 0, 1, 0, 1, ... that isperiodic, with period length 2. In this case the Poincar function is not a polynomial but rather an innite series1 +x2+x4+ which, being a geometric series, can be expressed as the rational function11 x2.More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above(e.g., a, b, c, a, b, c, . . . , has generating function(a +bx +cx2)/(1 x3)and more generally linear recursive sequences are exactly the sequences generated by rational functions; thus thePoincar series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursivesequence.The Poincar polynomials of the compact simple Lie groups are:PSU(n+1)(x) = (1 +x3)(1 +x5) (1 +x2n+1)PSO(2n+1)(x) = (1 +x3)(1 +x7) (1 +x4n1)PSp(n)(x) = (1 +x3)(1 +x7) (1 +x4n1)PSO(2n)(x) = (1 +x2n1)(1 +x3)(1 +x7) (1 +x4n5)PG2(x) = (1 +x3)(1 +x11)PF4(x) = (1 +x3)(1 +x11)(1 +x15)(1 +x23)PE6(x) = (1 +x3)(1 +x9)(1 +x11)(1 +x15)(1 +x17)(1 +x23)PE7(x) = (1 +x3)(1 +x11)(1 +x15)(1 +x19)(1 +x23)(1 +x27)(1 +x35)PE8(x) = (1 +x3)(1 +x15)(1 +x23)(1 +x27)(1 +x35)(1 +x39)(1 +x47)(1 +x59)3.6 Relationship with dimensions of spaces of dierential formsIn geometric situations when X is a closed manifold, the importance of the Betti numbers may arise from a dierentdirection, namely that they predict the dimensions of vector spaces of closed dierential forms modulo exact dif-ferential forms. The connection with the denition given above is via three basic results, de Rhams theorem andPoincar duality (when those apply), and the universal coecient theorem of homology theory.There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms. Thisrequires also the use of some of the results of Hodge theory, about the Hodge Laplacian.10 CHAPTER 3. BETTI NUMBERIn this setting, Morse theory gives a set of inequalities for alternating sums of Betti numbers in terms of a corre-sponding alternating sum of the number of critical points Ni of a Morse function of a given index:bi(X) bi1(X) + Ni Ni1 + .Witten gave an explanation of these inequalities by using the Morse function to modify the exterior derivative in thede Rham complex.[4]3.7 See alsoTopological data analysisTorsion coecient3.8 References[1] Barile, and Weisstein, Margherita and Eric. Betti number. From MathWorld--A Wolfram Web Resource.[2] Peter Robert Kotiuga (2010). A Celebration of the Mathematical Legacy of Raoul Bott. American Mathematical Soc. p.20. ISBN 978-0-8218-8381-5.[3] Per Hage (1996). Island Networks: Communication, Kinship, and Classication Structures in Oceania. Cambridge Univer-sity Press. p. 49. ISBN 978-0-521-55232-5.[4] Witten, Edward (1982). Supersymmetry and Morse theory. J. Dierential Geom. 17 (1982), no. 4, 661692.Warner, Frank Wilson (1983), Foundations of dierentiable manifolds and Lie groups, New York: Springer,ISBN 0-387-90894-3.Roe, John (1998), Elliptic Operators, Topology, and Asymptotic Methods, Research Notes in Mathematics Series395 (Second ed.), Boca Raton, FL: Chapman and Hall, ISBN 0-582-32502-1.Chapter 4Cauchy productIn mathematics, more specically in mathematical analysis, the Cauchy product is the discrete convolution of twosequences or two series. It is named after the French mathematician Augustin Louis Cauchy.4.1 DenitionsThe Cauchy product may apply to nite sequences,[1][2] innite sequences, nite series,[3] innite series,[4][5][6][7][8][9][10][11][12][13][14]power series,[15][16] etc. Convergence issues are discussed further down in the sections on Mertens theorem andCesros theorem.4.1.1 Cauchy product of two nite sequencesLet {ai} and {bj} be two nite sequences of complex numbers with the same length n. The Cauchy product of thesetwo nite sequences is equal to the Cauchy product of the nite seriesni=0ai andnj=0bj .4.1.2 Cauchy product of two innite sequencesLet {ai} and {bj} be two innite sequences of complex numbers. The Cauchy product of these two innite sequencesis equal to the Cauchy product of the innite seriesi=0ai andj=0bj .4.1.3 Cauchy product of two nite seriesLetni=0ai andnj=0bj be two nite series with complex terms. The Cauchy product of these two nite series isdened by a discrete convolution as follows:_ni=0ai___nj=0bj__ =nk=0ckwhere ck=kl=0albkl4.1.4 Cauchy product of two innite seriesLeti=0ai andj=0bj be two innite series with complex terms. The Cauchy product of these two innite seriesis dened by a discrete convolution as follows:_ i=0ai___j=0bj__ =k=0ckwhere ck=kl=0albkl1112 CHAPTER 4. CAUCHY PRODUCT4.1.5 Cauchy product of two power seriesConsider the following two power series with complex coecients {ai} and {bj} :i=0aixiandj=0bjxjThe Cauchy product of these two power series is dened by a discrete convolution as follows:_ i=0aixi___j=0bjxj__ =k=0ckxkwhere ck=kl=0albklIf these power series are formal power series, then we are manipulating series in disregard of any question ofconvergence: they need not be convergent series. Otherwise, see Mertens theorem and Cesros theorem belowfor convergence criteria.4.2 PropertyLetni=0ai andnj=0bj be two nite series with complex terms. The product of these two nite series satises theequation:_nk=0ak__nk=0bk_ =2nk=0ki=0aibki n1k=0_ak2nki=n+1bi +bk2nki=n+1ai_4.3 Convergence and Mertens theoremNot to be confused with Mertens theorems concerning distribution of prime numbers.Let (an)n and (bn)n be real or complex sequences. It was proved by Franz Mertens that, if the seriesn=0anconverges to A andn=0bn converges to B, and at least one of them converges absolutely, then their Cauchy productconverges to AB.It is not sucient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy productdoes not have to converge towards the product of the two series, as the following example shows:4.3.1 ExampleConsider the two alternating series withan= bn=(1)nn + 1,which are only conditionally convergent (the divergence of the series of the absolute values follows from the directcomparison test and the divergence of the harmonic series). The terms of their Cauchy product are given bycn=nk=0(1)kk + 1(1)nkn k + 1= (1)nnk=01(k + 1)(n k + 1)4.3. CONVERGENCE AND MERTENS THEOREM 13for every integer n 0. Since for every k {0, 1, ..., n} we have the inequalities k + 1 n + 1 and n k + 1 n +1, it follows for the square root in the denominator that (k + 1)(n k + 1) n +1, hence, because there are n + 1summands,|cn| nk=01n + 1 1for every integer n 0. Therefore, cn does not converge to zero as n , hence the series of the (cn)n divergesby the term test.4.3.2 Proof of Mertens theoremAssume without loss of generality that the seriesn=0an converges absolutely. Dene the partial sumsAn=ni=0ai, Bn=ni=0biand Cn=ni=0ciwithci=ik=0akbik .ThenCn=ni=0aniBiby rearrangement, henceFix > 0. Since kN|ak|< by absolute convergence, and since Bn converges to B as n , there exists aninteger N such that, for all integers n N,(this is the only place where the absolute convergence is used). Since the series of the (an)n converges, the individualan must converge to 0 by the term test. Hence there exists an integer M such that, for all integers n M,Also, since An converges to A as n , there exists an integer L such that, for all integers n L,Then, for all integers n max{L, M + N}, use the representation (1) for Cn, split the sumin two parts, use the triangleinequality for the absolute value, and nally use the three estimates (2), (3) and (4) to show that|Cn AB| =ni=0ani(Bi B) + (An A)BN1i=0|ani..M| |Bi B|. ./(3N)(3) by+ni=N|ani| |Bi B|. ./3(2) by+|An A| |B|. ./3(4) by .By the denition of convergence of a series, Cn AB as required.14 CHAPTER 4. CAUCHY PRODUCT4.4 Examples4.4.1 Finite seriesSuppose ai= 0 for all i > n and bi= 0 for all i > m. Here the Cauchy product ofan andbn is readily veriedto be (a0 + +an)(b0 + +bm) . Therefore, for nite series (which are nite sums), Cauchy multiplication isdirect multiplication of those series.4.4.2 Innite seriesFor some x, y R , let an= xn/n! and bn= yn/n! . Thencn=ni=0xii!yni(n i)!=1n!ni=0_ni_xiyni=(x +y)nn!by denition and the binomial formula. Since, formally, exp(x) =an and exp(y) =bn , we have shownthat exp(x + y) =cn . Since the limit of the Cauchy product of two absolutely convergent series is equal to theproduct of the limits of those series, we have proven the formula exp(x +y) = exp(x) exp(y) for all x, y R .As a second example, let an= bn= 1 for all n N . Then cn= n + 1 for all n N so the Cauchy productcn= (1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, . . . ) does not converge.4.5 Cesros theoremIn cases where the two sequences are convergent but not absolutely convergent, the Cauchy product is still Cesrosummable. Specically:If (an)n0 , (bn)n0 are real sequences withan A andbn B then1N_Nn=1ni=1ik=0akbik_ AB.This can be generalised to the case where the two sequences are not convergent but just Cesro summable:4.5.1 TheoremFor r> 1 and s> 1 , suppose the sequence (an)n0 is (C, r) summable with sum A and (bn)n0 is (C, s)summable with sum B. Then their Cauchy product is (C, r +s + 1) summable with sum AB.4.6 GeneralizationsAll of the foregoing applies to sequences in C (complex numbers). The Cauchy product can be dened for seriesin the Rnspaces (Euclidean spaces) where multiplication is the inner product. In this case, we have the result that iftwo series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.4.6.1 Products of nitely many innite seriesLet n N such that n 2 (actually the following is also true for n=1 but the statement becomes trivial in thatcase) and letk1=0a1,k1, . . . ,kn=0an,kn be innite series with complex coecients, from which all except then th one converge absolutely, and the n th one converges. Then the series4.7. RELATION TO CONVOLUTION OF FUNCTIONS 15k1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges and we have:k1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2=nj=1__kj=0aj,kj__This statement can be proven by induction over n : The case for n=2 is identical to the claim about the Cauchyproduct. This is our induction base.The induction step goes as follows: Let the claimbe true for an n Nsuch that n 2 , and letk1=0a1,k1, . . . ,kn+1=0an+1,kn+1be innite series with complex coecients, from which all except the n+1 th one converge absolutely, and the n+1th one converges. We rst apply the induction hypothesis to the series k1=0|a1,k1|, . . . ,kn=0|an,kn| . Weobtain that the seriesk1=0k1k2=0 kn1kn=0|a1,kna2,kn1kn an,k1k2|converges, and hence, by the triangle inequality and the sandwich criterion, the seriesk1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges, and hence the seriesk1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges absolutely. Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables,we have:n+1j=1__kj=0aj,kj__ =__kn+1=0=:akn+1..an+1,kn+1_________k1=0=:bk1 .. k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2_______=k1=0k1k2=0an+1,k1k2k2k3=0 knkn+1=0a1,kn+1a2,knkn+1 an,k2k3Therefore, the formula also holds for n + 1 .4.7 Relation to convolution of functionsOne can also dene the Cauchy product of doubly innite sequences, thought of as functions on Z . In this casethe Cauchy product is not always dened: for instance, the Cauchy product of the constant sequence 1 with itself,(. . . , 1, . . . ) is not dened. This doesn't arise for singly innite sequences, as these have only nite sums.One has some pairings, for instance the product of a nite sequence with any sequence, and the product 1 .This is related to duality of Lp spaces.16 CHAPTER 4. CAUCHY PRODUCT4.8 Notes[1] Dyer & Edmunds 2014, p. 190.[2] Weisstein, Cauchy Product.[3] Oberguggenberger & Ostermann 2011, p. 322.[4] Canuto & Tabacco 2015, p. 20.[5] Bloch 2011, p. 463.[6] Friedman & Kandel 2011, p. 204.[7] Ghorpade & Limaye 2006, p. 416.[8] Hijab 2011, p. 43.[9] Montesinos, Zizler & Zizler 2015, p. 98.[10] Oberguggenberger & Ostermann 2011, p. 322.[11] Pedersen 2015, p. 210.[12] Ponnusamy 2012, p. 200.[13] Pugh 2015, p. 210.[14] Sohrab 2014, p. 73.[15] Canuto & Tabacco 2015, p. 53.[16] Mathonline, Cauchy Product of Power Series.4.9 ReferencesApostol, Tom M. (1974), Mathematical Analysis (2nd ed.), Addison Wesley, p. 204, ISBN 978-0-201-00288-1.Bloch, Ethan D. (2011), The Real Numbers and Real Analysis, Springer.Canuto, Claudio; Tabacco, Anita (2015), Mathematical Analysis II (2nd ed.), Springer.Dyer, R.H.; Edmunds, D.E. (2014), From Real to Complex Analysis, Springer.Friedman, Menahem; Kandel, Abraham (2011), Calculus Light, Springer.Ghorpade, Sudhir R.; Limaye, Balmohan V. (2006), A Course in Calculus and Real Analysis, Springer.Hardy, G. H. (1949), Divergent Series, Oxford University Press, p. 227229.Hijab, Omar (2011), Introduction to Calculus and Classical Analysis (3rd ed.), Springer.Mathonline, Cauchy Product of Power Series.Montesinos, Vicente; Zizler, Peter; Zizler, Vclav (2015), An Introduction to Modern Analysis, Springer.Oberguggenberger, Michael; Ostermann, Alexander (2011), Analysis for Computer Scientists, Springer.Pedersen, Steen (2015), From Calculus to Analysis, Springer.4.9. REFERENCES 17Ponnusamy, S. (2012), Foundations of Mathematical Analysis, Birkhuser.Pugh, Charles C. (2015), Real Mathematical Analysis (2nd ed.), Springer.Sohrab, Houshang H. (2014), Basic Real Analysis (2nd ed.), Birkhuser.Weisstein, Eric W., Cauchy Product, From MathWorld--A Wolfram Web Resource.Chapter 5Cauchy sequence(a) The plot of a Cauchy sequence (xn), shown in blue, as xn versus n If the space containing the sequence is com-plete, the ultimate destination of this sequence (that is, the limit) exists.(b) A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as thesequence progresses.In mathematics, a Cauchy sequence (French pronunciation:[koi]; English pronunciation: /koi/ KOH-shee), namedafter Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequenceprogresses.[1] More precisely, given any small positive distance, all but a nite number of elements of the sequenceare less than that given distance from each other.The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences areknown to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, asopposed to the denition of convergence, which uses the limit value as well as the terms. This is often exploitedin algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce aCauchy sequence, consisting of the iterates, thus fullling a logical condition, such as termination.The notions above are not as unfamiliar as they might at rst appear. The customary acceptance of the fact that anyreal number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rationalnumbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In somecases it may be dicult to describe x independently of such a limiting process involving rational numbers.Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy lters and Cauchynets.185.1. IN REAL NUMBERS 195.1 In real numbersA sequencex1, x2, x3, . . .of real numbers is called a Cauchy sequence, if for every positive real number , there is a positive integer N suchthat for all natural numbers m, n > N|xm xn| < ,where the vertical bars denote the absolute value. In a similar way one can dene Cauchy sequences of rational orcomplex numbers. Cauchy formulated such a condition by requiring xm xn to be innitesimal for every pair ofinnite m, n.5.2 In a metric spaceTo dene Cauchy sequences in any metric space X, the absolute value |x - x| is replaced by the distance d(x, x)(where d : X X R with some specic properties, see Metric (mathematics)) between x and x.Formally, given a metric space (X, d), a sequencex1, x2, x3, ...is Cauchy, if for every positive real number > 0 there is a positive integer N such that for all positive integers m, n> N, the distanced(x, x) < .Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that thesequence ought to have a limit in X. Nonetheless, such a limit does not always exist within X.5.3 CompletenessA metric space X in which every Cauchy sequence converges to an element of X is called complete.5.3.1 ExamplesThe real numbers are complete under the metric induced by the usual absolute value, and one of the standardconstructions of the real numbers involves Cauchy sequences of rational numbers.A rather dierent type of example is aorded by a metric space X which has the discrete metric (where any twodistinct points are at distance 1 from each other). Any Cauchy sequence of elements of X must be constant beyondsome xed point, and converges to the eventually repeating term.5.3.2 Counter-example: rational numbersThe rational numbers Q are not complete (for the usual distance):There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having nolimit in Q. In fact, if a real number x is irrational, then the sequence (xn), whose n-th term is the truncation to ndecimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x.Irrational numbers certainly exist, for example:20 CHAPTER 5. CAUCHY SEQUENCEThe sequence dened by x0=1, xn+1=xn+2xn2consists of rational numbers (1, 3/2, 17/12,...), which isclear from the denition; however it converges to the irrational square root of two, see Babylonian method ofcomputing square root.The sequence xn=Fn/Fn1of ratios of consecutive Fibonacci numbers which, if it converges at all, con-verges to a limit satisfying 2= + 1 , and no rational number has this property. If one considers this as asequence of real numbers, however, it converges to the real number = (1+5)/2 , the Golden ratio, whichis irrational.The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational forany rational value of x0, but each can be dened as the limit of a rational Cauchy sequence, using, for instance,the Maclaurin series.5.3.3 Counter-example: open intervalThe open interval X = (0, 2) in the set of real numbers with an ordinary distance in R is not a complete space: thereis a sequence x = 1/n in it, which is Cauchy (for arbitrarily small distance bound d > 0 all terms x of n > 1/d t inthe (0, d) interval), however does not converge in X its 'limit', number 0, does not belong to the space X.5.3.4 Other propertiesEvery convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number > 0, beyondsome xed point, every term of the sequence is within distance /2 of s, so any two terms of the sequence arewithin distance of each other.Every Cauchy sequence of real (or complex) numbers is bounded (since for some N, all terms of the sequencefrom the N-th onwards are within distance 1 of each other, and if M is the largest absolute value of the termsup to and including the N-th, then no term of the sequence has absolute value greater than M+1).In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent(with the same limit), since, given any real number r > 0, beyond some xed point in the original sequence,every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are withindistance r/2 of each other, so every term of the original sequence is within distance r of s.These last two properties, together with a lemma used in the proof of the BolzanoWeierstrass theorem, yield onestandard proof of the completeness of the real numbers, closely related to both the BolzanoWeierstrass theoremand the HeineBorel theorem. The lemma in question states that every bounded sequence of real numbers has aconvergent monotonic subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has aconvergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness ofthe real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, ofconstructing the real numbers as the completion of the rational numbers, makes the completeness of the real numberstautological.One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use ofcompleteness is provided by consideration of the summation of an innite series of real numbers (or, more generally,of elements of any complete normed linear space, or Banach space). Such a series n=1xn is considered to beconvergent if and only if the sequence of partial sums (sm) is convergent, where sm=mn=1xn . It is a routinematter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q,sp sq=pn=q+1xn.If f :M N is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequencein M, then (f(xn)) is a Cauchy sequence in N. If (xn) and (yn) are two Cauchy sequences in the rational, real orcomplex numbers, then the sum (xn +yn) and the product (xnyn) are also Cauchy sequences.5.4. GENERALIZATIONS 215.4 Generalizations5.4.1 In topological vector spacesThere is also a concept of Cauchy sequence for a topological vector spaceX: Pick a local baseB forXabout0; then (xk) is a Cauchy sequence if for each memberV B , there is some numberNsuch that whenevern, m > N, xn xm is an element of V. If the topology of X is compatible with a translation-invariant metric d ,the two denitions agree.5.4.2 In topological groupsSince the topological vector space denition of Cauchy sequence requires only that there be a continuous subtractionoperation, it can just as well be stated in the context of a topological group: A sequence (xk) in a topological groupG is a Cauchy sequence if for every open neighbourhood U of the identity in G there exists some number N suchthat whenever m, n > N it follows that xnx1m U . As above, it is sucient to check this for the neighbourhoodsin any local base of the identity in G .As in the construction of the completion of a metric space, one can furthermore dene the binary relation on Cauchysequences in G that (xk) and (yk) are equivalent if for every open neighbourhood U of the identity in G there existssome number N such that whenever m, n > N it follows that xny1m U . This relation is an equivalence relation:It is reexive since the sequences are Cauchy sequences. It is symmetric sinceynx1m=(xmy1n)1U1which by continuity of the inverse is another open neighbourhood of the identity. It is transitive sincexnz1l=xny1mymz1l UU where U and U are open neighbourhoods of the identity such that UU U ; such pairsexist by the continuity of the group operation.5.4.3 In groupsThere is also a concept of Cauchy sequence in a groupG : LetH=(Hr) be a decreasing sequence of normalsubgroups of G of nite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H ) if and only if for any rthere is N such that m, n > N, xnx1m Hr .Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on G ,namely that for which H is a local base.The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences(s.th. r, N, n > N, xn Hr ) is a normal subgroup of C . The factor group C/C0 is called the completion ofG with respect to H .One can then show that this completion is isomorphic to the inverse limit of the sequence (G/Hr) .An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adiccompletion of the integers with respect to a prime p. In this case, G is the integers under addition, and Hr is theadditive subgroup consisting of integer multiples of pr.IfHis a conal sequence (i.e., any normal subgroup of nite index contains someHr), then this completion iscanonical in the sense that it is isomorphic to the inverse limit of (G/H)H , where H varies over all normal subgroupsof nite index. For further details, see ch. I.10 in Lang's Algebra.5.4.4 In constructive mathematicsIn constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to beuseful. If (x1, x2, x3, ...) is a Cauchy sequence in the set X , then a modulus of Cauchy convergence for the sequenceis a function from the set of natural numbers to itself, such that km, n > (k), |xm xn| < 1/k .Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchysequence has a modulus) follows from the well-ordering property of the natural numbers (let (k) be the smallestpossible Nin the denition of Cauchy sequence, taking r to be 1/k ). However, this well-ordering property doesnot hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, thisconverse also follows (directly) from the principle of dependent choice (in fact, it will follow from the weaker AC00),22 CHAPTER 5. CAUCHY SEQUENCEwhich is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directlyonly by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.That said, using a modulus of Cauchy convergence can simplify both denitions and theorems in constructive analysis.Perhaps even more useful are regular Cauchy sequences, sequences with a given modulus of Cauchy convergence(usually (k)=k or (k)=2k). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (inthe sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved withoutusing any form of the axiom of choice. Regular Cauchy sequences were used by Errett Bishop in his Foundationsof Constructive Analysis, but they have also been used by Douglas Bridges in a non-constructive textbook (ISBN978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread faroutside of that milieu.5.4.5 In a hyperreal continuumA real sequence un: n N has a natural hyperreal extension, dened for hypernatural values H of the index n inaddition to the usual natural n. The sequence is Cauchy if and only if for every innite H and K, the values uH anduK are innitely close, or adequal, i.e.st(uH uK) = 0where st is the standard part function.5.5 See alsoModes of convergence (annotated index)5.6 References[1] Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl0848.130015.7 Further readingBourbaki, Nicolas (1972). Commutative Algebra (English translation ed.). Addison-Wesley. ISBN 0-201-00644-8.Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN978-0-201-55540-0, Zbl 0848.13001Spivak, Michael (1994). Calculus (3rd ed.). Berkeley, CA: Publish or Perish. ISBN 0-914098-89-6.Troelstra, A. S.; D. van Dalen. Constructivism in Mathematics: An Introduction. (for uses in constructivemathematics)5.8 External linksHazewinkel, Michiel, ed. (2001), Fundamental sequence, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4Chapter 6Chebyshevs sum inequalityFor the similarly named inequality in probability theory, see Chebyshevs inequality.In mathematics, Chebyshevs sum inequality, named after Pafnuty Chebyshev, states that ifa1 a2 anandb1 b2 bn,then1nnk=1ak bk _1nnk=1ak__1nnk=1bk_.Similarly, ifa1 a2 anandb1 b2 bn,then1nnk=1akbk _1nnk=1ak_ _1nnk=1bk_. [1]6.1 ProofConsider the sumS=nj=1nk=1(aj ak)(bj bk).2324 CHAPTER 6. CHEBYSHEVS SUM INEQUALITYThe two sequences are non-increasing, therefore aj ak and bj bk have the same sign for any j, k. Hence S 0.Opening the brackets, we deduce:0 2nnj=1ajbj 2nj=1ajnk=1bk,whence1nnj=1ajbj __1nnj=1aj____1nnj=1bk__.An alternative proof is simply obtained with the rearrangement inequality.6.2 Continuous versionThere is also a continuous version of Chebyshevs sum inequality:If f and g are real-valued, integrable functions over [0,1], both non-increasing or both non-decreasing, then10f(x)g(x)dx 10f(x)dx10g(x)dx,with the inequality reversed if one is non-increasing and the other is non-decreasing.6.3 Notes[1] Hardy, G. H.; Littlewood, J. E.; Plya, G. (1988). Inequalities. Cambridge Mathematical Library. Cambridge: CambridgeUniversity Press. ISBN 0-521-35880-9. MR 0944909.Chapter 7Complementary sequencesFor complementary sequences in biology, see complementarity (molecular biology).In applied mathematics, complementary sequences (CS) are pairs of sequences with the useful property that theirout-of-phase aperiodic autocorrelation coecients sum to zero. Binary complementary sequences were rst intro-duced by Marcel J. E. Golay in 1949. In 19611962 Golay gave several methods for constructing sequences of length2Nand gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method forconstructing sequences of length mn from sequences of lengths m and n which allows the construction of sequencesof any length of the form 2N10K26M.Later the theory of complementary sequences was generalized by other authors t