Famous mathematical constants include the ratio of circular circumference to diameter, π = 3.14 ... , and the natural logarithm base, e = 2.718 ... Students and professionals can often name a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here the author renews the search he began in his book Mathematical Constants, adding another 133 essays that broaden the landscape. Topics include the minimality of soap film surfaces, prime numbers, elliptic curves and modular forms, Poisson–Voronoi tessellations, random triangles, Brownian motion, uncertainty inequalities, Prandtl–Blasius flow (from fluid dynamics), Lyapunov exponents, knots and tangles, continued fractions, Galton–Watson trees, electrical capacitance (from potential theory), Zermelo's navigation problem, and the optimal control of a pendulum. Unsolved problems appear virtually everywhere as well. This volume continues an outstanding scholarly attempt to bring together all significant mathematical constants in one place.
Author(s): Steven R. Finch
Series: Encyclopedia Of Mathematics And Its Applications Vol. 169
Publisher: Cambridge University Press
Year: 2018
Language: English
Pages: 782
Tags: Mathematical Constants
Contents......Page 6
Preface......Page 10
Notation......Page 12
1.1 Bipartite, k-Colorable and k-Colored Graphs......Page 14
1.2 Transitive Relations, Topologies and Partial Orders......Page 20
1.3 Series-Parallel Networks......Page 25
1.4 Two Asymptotic Series......Page 34
1.5 Multiples and Divisors......Page 45
1.6 Discrepancy and Uniformity......Page 53
1.7 Unitarism and Infinitarism......Page 62
1.8 Erdős’ Minimum Overlap Problem......Page 69
1.9 Planar Graph Growth Constants......Page 71
1.10 Tauberian Constants......Page 80
1.11 Integer Partitions......Page 85
1.12 Class Number Theory......Page 91
1.13 Quadratic Dirichlet L-Series......Page 110
1.14 Elliptic Curves over Q......Page 125
1.15 Modular Forms on SL2(Z)......Page 145
1.16 Chebyshev’s Bias......Page 160
1.17 Pattern-Avoiding Permutations......Page 162
1.18 Cyclic Group Orders......Page 167
1.19 Dedekind Eta Products......Page 170
1.20 Series involving Arithmetric Functions......Page 173
1.21 Riemann Zeta Moments......Page 189
1.22 Central Binomial Coefficients......Page 195
1.23 Fractional Parts of Bernoulli Numbers......Page 202
1.24 Products of Consecutive-Integer Ratios......Page 205
1.25 Prime Number Theorem......Page 208
1.26 Mertens’ Formula......Page 215
1.27 Cyclotomic Polynomials......Page 219
1.28 Minkowski–Alkauskas Constant......Page 220
1.29 Two-Colorings of Positive Integers......Page 222
1.30 Signum Equations and Extremal Coefficients......Page 226
1.31 Monoids of Natural Numbers......Page 230
1.32 Primitive Cusp Form......Page 236
1.33 Cubic and Quartic Characters......Page 245
1.34 Distribution of Error Terms......Page 253
1.35 Cilleruelo’s LCM Constants......Page 255
1.36 Amicable Pairs and Aliquot Sequences......Page 257
1.37 Fermat Numbers and Elite Primes......Page 260
1.38 Average Least Nonresidues......Page 262
1.39 Apollonian Circles with Integer Curvatures......Page 266
1.40 Molteni’s Composition Constant......Page 271
1.41 Boolean Decision Functions......Page 273
1.42 Map Asymptotics Constant......Page 276
1.43 Injections, Surjections and More......Page 279
2.1 Hardy–Littlewood Maximal Inequalities......Page 285
2.2 Bessel Function Zeroes......Page 288
2.3 Nash’s Inequality......Page 297
2.4 Uncertainty Inequalities......Page 304
2.5 Airy Function Zeroes......Page 308
2.6 Projections of Minimal Norm......Page 312
2.7 Bohr’s Inequality......Page 316
2.8 Moduli of Continuity......Page 319
2.9 Quinn–Rand–Strogatz Constant......Page 324
2.10 Tsirelson’s Constant......Page 327
2.11 Mathieu Eigenvalues......Page 334
2.12 Thomas–Fermi Model......Page 337
2.13 Prandtl–Blasius Flow......Page 342
2.14 Lane–Ritter–Emden Constants......Page 351
2.15 Radiative Transfer Equations......Page 357
2.16 Carleman’s Inequality......Page 365
2.17 Golay–Littlewood Problem......Page 367
2.18 Online Matching Coins......Page 371
2.19 Toothpicks and Live Cells......Page 376
2.20 Virial Coefficients......Page 378
2.21 Strong Triangle Inequality......Page 386
3.1 Radii in Geometric Function Theory......Page 390
3.2 Numerical Radii of Linear Operators......Page 400
3.3 Coefficient Estimates for Univalent Functions......Page 403
3.4 Planar Harmonic Mappings......Page 412
3.5 Constant of Interpolation......Page 416
3.6 Dirichlet Integral......Page 419
3.7 Brachistochrone Problem......Page 420
3.8 Unconditional Basis Constants......Page 423
3.9 Power Series with Restricted Coefficients......Page 427
3.10 Hankel and Toeplitz Determinants......Page 428
3.11 Gol dberg’s Zero-One Constants......Page 432
3.12 Electrical Capacitance......Page 436
3.13 Aissen’s Convex Set Function......Page 441
3.14 Condition Numbers of Matrices......Page 443
3.15 Goddard’s Rocket Problem......Page 445
3.16 Swing-Up Control of a Pendulum......Page 450
3.17 Zermelo’s Navigation Problem......Page 457
4.1 Hammersley’s Path Process......Page 468
4.2 Moments of Sums......Page 475
4.3 Ornstein–Uhlenbeck Process......Page 481
4.4 Zero Crossings......Page 492
4.5 Variants of Brownian Motion......Page 498
4.6 Shapes of Binary Trees......Page 507
4.7 Expected Lifetimes and Inradii......Page 517
4.8 Subcritical Galton–Watson Trees......Page 524
4.9 Continued Fraction Transformation......Page 532
4.10 Continued Fraction Transformation. II......Page 540
4.11 Continued Fraction Transformation. III......Page 546
4.12 Continued Fraction Transformation. IV......Page 554
4.13 Lyapunov Exponents......Page 565
4.14 Lyapunov Exponents. II......Page 571
4.15 Lyapunov Exponents. III......Page 577
4.16 Lyapunov Exponents. IV......Page 581
4.17 Stars and Watermelons......Page 590
4.18 Prophet Inequalities......Page 593
4.19 Excursion Durations......Page 595
4.20 Gambler’s Ruin......Page 597
4.21 Self-Convolutions......Page 602
4.22 Newcomb–Benford Law......Page 605
4.23 Electing a Leader......Page 607
4.24 Substitution Dynamics......Page 612
4.25 Biham–Middleton–Levine Traffic......Page 617
4.26 Contact Processes......Page 618
4.27 Interpolating between Max and Sum......Page 625
4.28 Mixing Time of Markov Chains......Page 629
4.29 Correlated Products......Page 632
5.1 Knots, Links and Tangles......Page 636
5.2 Convex Lattice Polygons......Page 645
5.3 Volumes of Hyperbolic 3-Manifolds......Page 649
5.4 Poisson–Voronoi Tessellations......Page 656
5.5 Optimal Escape Paths......Page 665
5.6 Minkowski–Siegel Mass Constants......Page 667
5.7 Slicing Problem......Page 673
5.8 Constant of Theodorus......Page 675
5.9 Nearest-Neighbor Graphs......Page 677
5.10 Random Triangles......Page 681
5.11 Random Triangles. II......Page 695
5.12 Random Triangles. III......Page 702
5.13 Random Triangles. IV......Page 708
5.14 Random Triangles. V......Page 713
5.15 Random Triangles. VI......Page 726
5.16 Colliding Dice Probabilities......Page 731
5.17 Gergonne–Schwarz Surface......Page 734
5.18 Partitioning Problem......Page 744
5.19 Soap Film Experiments......Page 753
5.20 Inflating an Inelastic Membrane......Page 760
5.21 Enumerative Geometry......Page 763
5.22 Distance-Avoiding Sets in the Plane......Page 767
5.23 Fraenkel Asymmetry......Page 770
Index......Page 778
