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Addition chainsFrom Wikipedia, the free encyclopediaContents1 Addition chain 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Methods for computing addition chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Chain length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Brauer chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Scholz conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Addition-chain exponentiation 42.1 Addition-subtractionchain exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Addition-subtraction chain 63.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Almost convergent sequence 84.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Arithmetic progression 95.1 Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Betti number 136.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Example 1: Betti numbers of a simplicial complex K. . . . . . . . . . . . . . . . . . . . . . . . . 14iii CONTENTS6.3 Example 2: the rst Betti number in graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.6 Relationship with dimensions of spaces of dierential forms . . . . . . . . . . . . . . . . . . . . . 166.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Cauchy product 187.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.1 Cauchy product of two nite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.2 Cauchy product of two innite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.3 Cauchy product of two nite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.4 Cauchy product of two innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.1.5 Cauchy product of two power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Convergence and Mertens theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.2 Proof of Mertens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.4.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.4.2 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.5 Cesros theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.5.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6.1 Products of nitely many innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 Relation to convolution of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Cauchy sequence 258.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.1 In topological vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28CONTENTS iii8.4.5 In a hyperreal continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Chebyshevs sum inequality 309.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Continuous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110Complementary sequences 3210.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.3Properties of complementary pairs of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.4Golay pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.5Applications of complementary sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511Cumulant-generating function 3611.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.1.1 Alternative denition of the cumulant generating function . . . . . . . . . . . . . . . . . . 3611.2Uses in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3Cumulants of some discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . 3711.4Cumulants of some continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . . 3811.5Some properties of the cumulant generating function . . . . . . . . . . . . . . . . . . . . . . . . . 3811.6Some properties of cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6.1 Invariance and equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6.2 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6.3 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6.4 A negative result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6.5 Cumulants and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.6.6 Relation to moment-generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6.7 Cumulants and set-partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.6.8 Cumulants and combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.7Joint cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.7.1 Conditional cumulants and the law of total cumulance . . . . . . . . . . . . . . . . . . . . 4411.8Relation to statistical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.9History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.10Cumulants in generalized settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.10.1 Formal cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45iv CONTENTS11.10.2 Bell numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.10.3 Cumulants of a polynomial sequence of binomial type . . . . . . . . . . . . . . . . . . . . 4611.10.4 Free cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712Cutting sequence 4812.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813Cyclic sieving 4913.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.3Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014DavenportSchinzel sequence 5114.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.2Length bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.3Application to lower envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415Disjunctive sequence 5515.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.2Rich numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716Divisibility sequence 5816.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817Ducci sequence 6017.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.3Modulo two form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.4Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.5Other related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218Examples of generating functions 63CONTENTS v18.1Worked example A: basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.1.1 Bivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.2Worked example B: Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419Factorial moment generating function 6519.1Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620Farey sequence 6720.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.3.1 Sequence length and index of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.3.2 Farey neighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.3.4 Ford circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.3.5 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.4Next term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521Generating function 7621.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.1.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.1.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.1.3 Poisson generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.1.4 Lambert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.1.5 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.1.6 Dirichlet series generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.1.7 Polynomial sequence generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.2Ordinary generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.2.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7921.2.2 Multiplication yields convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8021.2.3 Relation to discrete-time Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 8021.2.4 Asymptotic growth of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8021.2.5 Bivariate and multivariate generating functions . . . . . . . . . . . . . . . . . . . . . . . 8121.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8121.3.1 Ordinary generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.3.2 Exponential generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82vi CONTENTS21.3.3 Bell series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.3.4 Dirichlet series generating function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.3.5 Multivariate generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221.4.1 Techniques of evaluating sums with generating function . . . . . . . . . . . . . . . . . . . 8321.4.2 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8321.4.3 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.5Other generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622Geometric progression 8722.1Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8822.2Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8922.2.3 Innite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9022.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.3Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.4Relationship to geometry and Euclids work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9322.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9422.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9423Halton sequence 9523.1Example of Halton sequence used to generate points in (0, 1) (0, 1) in R2. . . . . . . . . . . . . 9523.2Implementation in Pseudo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9723.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9724Harmonic progression (mathematics) 9824.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.2Use in geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925Innite product 10025.1Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.2Product representations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101CONTENTS vii25.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10225.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10226Interleave sequence 10326.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327Iterated function 10427.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.2Abelian property and Iteration sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10427.3Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.4Limiting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.5Fractional iterates and ows, and negative iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527.6Some formulas for fractional iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10627.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10627.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10627.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.7Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.8Markov chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.9Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.10Means of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.11In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.12Denitions in terms of iterated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.13Lies data transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10827.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10927.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10928Katydid sequence 11028.1Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11028.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11028.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029Limit of a sequence 11129.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.2Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11229.2.2 Formal Denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11329.3Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.4Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114viii CONTENTS29.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.5Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.6Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11429.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.8.1 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11529.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11629.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11630List of sums of reciprocals 11730.1Finitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11730.2Innitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.2.1 Convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.2.2 Divergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11931Logarithmically concave sequence 12031.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12031.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12032Low-discrepancy sequence 12132.1Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12132.1.1 Low-discrepancy sequences in numerical integration . . . . . . . . . . . . . . . . . . . . 12132.2Denition of discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12232.3The KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12332.4The formula of Hlawka-Zaremba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12332.5The L2version of the KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 12332.6The ErdsTurnKoksma inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.7The main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.8Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12532.9Construction of low-discrepancy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12532.9.1 Random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12532.9.2 Additive recurrence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.9.3 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12632.9.4 van der Corput sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.9.5 Halton sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.9.6 Hammersley set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12732.9.7 Poisson disk sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.10Graphical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128CONTENTS ix32.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12833Mathematics of oscillation 13633.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13633.1.1 Oscillation of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13633.1.2 Oscillation of a function on an open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13733.1.3 Oscillation of a function at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13733.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13733.3Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13733.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13933.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13934Matsushimas formula 14034.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14035Moment-generating function 14135.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14135.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14235.3Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14235.3.1 Sum of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14235.3.2 Vector-valued random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14235.4Important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.4.1 Calculations of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.5Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.6Relation to other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14335.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14435.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14436Monotone convergence theorem 14536.1Convergence of a monotone sequence of real numbers . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.3 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.1.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14536.2Convergence of a monotone series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636.3Lebesgues monotone convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636.3.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14636.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14736.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149x CONTENTS36.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14937Periodic sequence 15037.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.3Periodic 0, 1 sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15137.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15138Polynomial sequence 15238.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238.2Classes of polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15339Polyphase sequence 15439.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15439.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15440Probability-generating function 15540.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.1.1 Univariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.1.2 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.2.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15540.2.2 Probabilities and expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.2.3 Functions of independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 15640.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15740.4Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15840.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15840.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15841Random sequence 15941.1Early history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041.2Modern approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142Rook polynomial 16242.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16242.1.1 Complete boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.2Matching polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.3Connection to matrix permanents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163CONTENTS xi42.4Complete rectangular boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.4.1 Rooks problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.4.2 The rook polynomial as a generalization of the rooks problem. . . . . . . . . . . . . . . . 16442.4.3 Symmetric arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.4.4 Arrangements counted by symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . 16642.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16643Scholz conjecture 16843.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16843.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16844Sequence 16944.1Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17044.1.1 Important examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17044.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17144.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17244.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.2.3 Increasing and decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17344.3Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17444.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17544.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.4Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17644.5Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.5.2 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17744.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.5.5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.6Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.7Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.8Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181xii CONTENTS45Sequence space 18245.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.1.1 pspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18245.1.2 c and c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345.1.3 Other sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18345.2Properties of pspaces and the space c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.2.1 pspaces are increasing in p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18546Shift rule 18646.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647Sobol sequence 18747.1Good distributions in the s-dimensional unit hypercube . . . . . . . . . . . . . . . . . . . . . . . 18747.2A fast algorithm for the construction of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . 18847.3Additional uniformity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18847.4The initialization of Sobol numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.5Implementation and availability of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . . . 18947.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19048Stationary sequence 19148.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19148.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19149Sturmian word 19249.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19249.1.1 Combinatoric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19249.1.2 Geometric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.2Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19349.2.2 Balanced aperiodic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.2.3 Slope and intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19449.2.4 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.3Non-binary words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.4Associated real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.5History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19549.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19649.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196CONTENTS xiii50Subadditivity 19750.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19750.3Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19951Subsequence 20051.1Common subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20051.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20051.3Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20151.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20151.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20152Subsequential limit 20253Superadditivity 20353.1Examples of superadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20353.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20353.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20454Tuple 20554.1Etymology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.1.1 Names for tuples of specic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20554.3Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.3.1 Tuples as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.3.2 Tuples as nested ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.3.3 Tuples as nested sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20754.4 n-tuples of m-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20754.5Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20754.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20854.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20854.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20955Van der Corput sequence 21055.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21155.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21155.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21155.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21156Vectorial addition chain 212xiv CONTENTS56.1Addition sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21256.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21356.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21357Vites formula 21457.1Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21557.2Interpretation and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21557.3Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21657.4Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21657.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21757.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21858Weisners method 21958.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21958.2Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 22058.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22058.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22458.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Chapter 1Addition chainIn mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbersv and a sequence of index pairs w such that each term in v is the sum of two previous terms, the indices of thoseterms being specied by w:v =(v0,...,vs), with v0 = 1 and vs = nfor each 0< i s holds: vi = vj + vk, with wi=(j,k) and 0 j,k i 1Often only v is given since it is easy to extract w from v, but sometimes w is not uniquely reconstructible. Anintroduction is given by Knuth.[1]1.1 ExamplesAs an example: v = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since2 = 1 + 13 = 2 + 16 = 3 + 312 = 6 + 624 = 12 + 1230 = 24 + 631 = 30 + 1Addition chains can be used for addition-chain exponentiation: so for example we only need 7 multiplications tocalculate 531:52= 51 5153= 52 5156= 53 53512= 56 56524= 512 512530= 524 56531= 530 5112 CHAPTER 1. ADDITION CHAIN1.2 Methods for computing addition chainsCalculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one mustnd a chain that simultaneously forms each of a sequence of values, is NP-complete.[2] There is no known algorithmwhich can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or smallmemory usage. However, several techniques to calculate relatively short chains exist. One very well known techniqueto calculate relatively short addition chains is the binary method, similar to exponentiation by squaring. Other well-known methods are the factor method and window method.[3]1.3 Chain lengthLet l(n) denote the smallest s so that there exists an addition chain of length s which computes n. It is known that [4]log2(n) + log2((n)) 2.13 l(n) log2(n) + log2(n)(1 +o(1))/ log2(log2(n))where (n) is Hamming weight of binary expansion of n.It is clear that l(2n) l(n)+1. Strict inequality is possible, as l(382) = l(191) = 11, observed by Knuth.[5] The rstinteger with l(2n) < l(n) is n = 375494703.[6]1.4 Brauer chainA Brauer chain or star addition chain is an addition chain in which one of the summands is always the previouschain: that is,for each k>0: ak = ak-1 + aj for some j < k.A Brauer number is one for which the Brauer chain is minimal.[5]Brauer proved thatl*(2n1) n 1 + l*(n)where l* is the length of the shortest star chain. For many values of n,and in particular for n 2500, they are equal:l(n) = l*(n). But Hansen showed that there are some values of n for which l(n) l*(n), such as n = 26106+ 23048+22032+ 22016+ 1 which has l*(n) = 6110, l(n) 6109.1.5 Scholz conjectureMain article: Scholz conjectureThe Scholz conjecture (sometimes called the ScholzBrauer or BrauerScholz conjecture), named after A. Scholz andAlfred T. Brauer), is a conjecture from 1937 stating thatl(2n 1) n 1 + l(n) .It is known to be true for Hansen numbers, a generalization of Brauer numbers; N. Clift checked by computer thatall n5784688 are Hansen (while 5784689 is not).[6] Clift further checked that is true with equality for n64.[5]1.6. SEE ALSO 31.6 See alsoAddition chain exponentiationAddition-subtraction chainVectorial addition chainLucas chain1.7 References[1] D. E. Knuth, The Art of Computer Programming, Vol 2, Seminumerical Algorithms, Section 4.6.3, 3rd edition, 1997[2] Downey, Peter; Leong, Benton; Sethi, Ravi (1981). Computing sequences with addition chains. SIAM Journal onComputing 10 (3): 638646. doi:10.1137/0210047.. A number of other papers state that nding a single addition chain isNP-complete, citing this paper, but it does not claim or prove such a result.[3] Otto, Martin (2001), Brauer addition-subtraction chains (PDF), Diplomarbeit, University of Paderborn.[4] A. Schnhage A lower bound on the length of addition chains, Theoret. Comput. Sci. 1 (1975), 112.[5] Guy (2004) p.169[6] Clift, Neill Michael (2011). Calculating optimal addition chains (PDF). Computing 91 (3): 265284. doi:10.1007/s00607-010-0118-8.Brauer, Alfred (1939). On addition chains. Bulletin of the American Mathematical Society 45 (10): 736739.doi:10.1090/S0002-9904-1939-07068-7. ISSN 0002-9904. MR 0000245.Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7.OCLC 54611248. Zbl 1058.11001. Section C6.1.8 External linkshttp://wwwhomes.uni-bielefeld.de/achim/addition_chain.html"Sloanes A003313 : Length of shortest addition chain for n", The On-Line Encyclopedia of Integer Sequences.OEIS Foundation.F. Bergeron, J. Berstel. S. Brlek Ecient computation of addition chainsChapter 2Addition-chain exponentiationIn mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation bypositive integer powers that requires a minimal number of multiplications. It works by creating the shortest additionchain that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying two ofthe earlier exponentiation results. More generally, addition-chain exponentiation may also refer to exponentiation bynon-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very dicult tond).The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less.The rst example of where it does better is for a15, where the binary method needs six multiplications but a shortestaddition chain requires only ve:a15= a (a [a a2]2)2a15= a3([a3]2)2On the other hand, the determination of a shortest addition chain is hard: no ecient optimal methods are currentlyknown for arbitrary exponents, and the related problemof nding a shortest addition chain for a given set of exponentshas been proven NP-complete.[1] Even given a shortest chain, addition-chain exponentiation requires more memorythan the binary method, because it must potentially store many previous exponents from the chain. So in practice,shortest addition-chain exponentiation is primarily used for small xed exponents for which a shortest chain can beprecomputed and is not too large.There are also several methods to approximate a shortest addition chain, and which often require fewer multiplica-tions than binary exponentiation; binary exponentiation itself is a suboptimal addition-chain algorithm. The optimalalgorithm choice depends on the context (such as the relative cost of the multiplication and the number of times agiven exponent is re-used).[2]The problem of nding the shortest addition chain cannot be solved by dynamic programming, because it does notsatisfy the assumption of optimal substructure. That is, it is not sucient to decompose the power into smallerpowers, each of which is computed minimally, since the addition chains for the smaller powers may be related (toshare computations). For example, in the shortest addition chain for a15above, the subproblem for a6must becomputed as (a3)2since a3is re-used (as opposed to, say, a6= a2(a2)2, which also requires three multiplies).2.1 Addition-subtractionchain exponentiationIf both multiplication and division are allowed, then an addition-subtraction chain may be used to obtain even fewertotal multiplications+divisions (where subtraction corresponds to division). However, the slowness of division com-pared to multiplication makes this technique unattractive in general. For exponentiation to negative integer powers,on the other hand, since one division is required anyway, an addition-subtraction chain is often benecial. Onesuch example is a31, where computing 1/a31by a shortest addition chain for a31requires 7 multiplications and onedivision, whereas the shortest addition-subtraction chain requires 5 multiplications and one division:42.2. REFERENCES 5a31= a/((((a2)2)2)2)2For exponentiation on elliptic curves, the inverse of a point (x, y) is available at no cost, since it is simply (x, y), andtherefore addition-subtraction chains are optimal in this context even for positive integer exponents.[3]2.2 References[1] Downey, Peter; Leong, Benton; Sethi, Ravi (1981). Computing sequences with addition chains. SIAM Journal onComputing 10 (3): 638646. doi:10.1137/0210047.[2] Gordon, D. M. (1998). Asurvey of fast exponentiation methods (PDF). J. Algorithms 27: 129146. doi:10.1006/jagm.1997.0913.[3] Franois Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtraction chains,RAIRO Informatique thoretique et application 24, pp. 531-543 (1990).Donald E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition, 4.6.3(Addison-Wesley: San Francisco, 1998).Daniel J. Bernstein, "Pippengers Algorithm, to be incorporated into authors High-speed cryptography book.(2002)Chapter 3Addition-subtraction chainAn addition-subtraction chain, a generalization of addition chains to include subtraction, is a sequence a0, a1, a2,a3, ... that satisesa0= 1,fork > 0, ak= ai aj some for 0 i, j< k.An addition-subtraction chain for n, of length L, is an addition-subtraction chain such that aL= n . That is, one canthereby compute n by L additions and/or subtractions. (Note that n need not be positive. In this case, one may alsoinclude a=0 in the sequence, so that n=1 can be obtained by a chain of length 1.)By denition, every addition chain is also an addition-subtraction chain, but not vice versa. Therefore, the lengthof the shortest addition-subtraction chain for n is bounded above by the length of the shortest addition chain for n.In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining aminimum addition chain) is a dicult problem for which no ecient algorithms are currently known. The relatedproblemof nding an optimal addition sequence is NP-complete (Downey et al., 1981), but it is not known for certainwhether nding optimal addition or addition-subtraction chains is NP-hard.For example, one addition-subtraction chain is:a0= 1 , a1= 2 = 1 + 1 , a2= 4 = 2 + 2 , a3= 3 = 4 1 . Thisis not a minimal addition-subtraction chain for n=3, however, because we could instead have chosen a2= 3 = 2 +1. The smallest n for which an addition-subtraction chain is shorter than the minimal addition chain is n=31, whichcan be computed in only 6 additions (rather than 7 for the minimal addition chain):a0= 1,a1= 2 = 1+1,a2= 4 = 2+2,a3= 8 = 4+4,a4= 16 = 8+8,a5= 32 = 16+16,a6= 31 = 321.Likeanadditionchain, anaddition-subtractionchaincanbeusedforaddition-chainexponentiation: giventheaddition-subtraction chain of length L for n, the power xncan be computed by multiplying or dividing by x L times,where the subtractions correspond to divisions. This is potentially ecient in problems where division is an inexpen-sive operation, most notably for exponentiation on elliptic curves where division corresponds to a mere sign change(as proposed by Morain and Olivos, 1990).Some hardware multipliers multiply by n using an addition chain described by n in binary:n = 31 = 0 0 0 1 1 1 1 1 (binary).Other hardware multipliers multiply by n using an addition-subtraction chain described by n in Booth encoding:n = 31 = 0 0 1 0 0 0 0 1 (Booth encoding).3.1 ReferencesHugo Volger, Some results on addition/subtraction chains, Information Processing Letters 20, pp. 155160(1985).63.1. REFERENCES 7Franois Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtractionchains, RAIRO Informatique thoretique et application 24, pp. 531543 (1990).Peter Downey, Benton Leong, and Ravi Sethi, Computing sequences with addition chains, SIAMJ. Computing10 (3), 638-646 (1981).Sequence A128998, length of minimum addition-subtraction chain, The On-Line Encyclopedia of IntegerSequences.Chapter 4Almost 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.4.1 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.8Chapter 5Arithmetic 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.5.1 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)2910 CHAPTER 5. ARITHMETIC PROGRESSIONIn 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= 32.5.1.1 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).5.2 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 product5.3. STANDARD DEVIATION 11m(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 1098.5.3 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 terms5.4 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.5.5 Formulas at a GlanceIfa1andnSnnthenan= a1 + (n 1)d,an= am + (n m)d.Sn=n2[2a1 + (n 1)d].Sn=n(a1 +an)212 CHAPTER 5. ARITHMETIC PROGRESSION5.n = Sn/nn =a1 +an2.5.6 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 progressionsUtonality5.7 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 0-387-95419-8.5.8 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 6Betti 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.6.1 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 simplicial1314 CHAPTER 6. BETTI NUMBERcomplexes: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).6.2 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 +x6.3 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 componentsequals6.4. PROPERTIES 15mn +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]6.4 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).6.5 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, ... .16 CHAPTER 6. BETTI NUMBERthe 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)6.6 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.6.7. SEE ALSO 17In 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]6.7 See alsoTopological data analysisTorsion coecient6.8 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 7Cauchy 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.7.1 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.7.1.1 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 .7.1.2 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 .7.1.3 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=0albkl7.1.4 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=0albkl187.2. PROPERTY 197.1.5 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.7.2 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_7.3 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:7.3.1 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)20 CHAPTER 7. CAUCHY PRODUCTfor 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.7.3.2 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.7.4. EXAMPLES 217.4 Examples7.4.1 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.7.4.2 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.7.5 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:7.5.1 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.7.6 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.7.6.1 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 series22 CHAPTER 7. CAUCHY PRODUCTk1=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 .7.7 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.7.8. NOTES 237.8 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.7.9 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.24 CHAPTER 7. CAUCHY PRODUCTPonnusamy, 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 8Cauchy 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.2526 CHAPTER 8. CAUCHY SEQUENCE8.1 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.8.2 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