As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83 , then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]] k ] is equal for i = 0 and i = 1 if s is a sequence of the form (b) on page 83 with length 2 m , m > k .

Quantum effects Over the years, many suggested effects have been thought to be characteristic of quantum systems: • Basic quantization (1913): mechanical properties of particles in effectively bounded systems are discrete; • Wave-particle duality (1923): objects like electrons and photons can be described as either waves or particles; • Spin (1925): particles can have intrinsic angular momentum even if they are of zero size; • Non-commuting measurements (1926): one can get different results doing measurements in different orders; • Complex amplitudes (1926): processes are described by complex probability amplitudes; • Probabilism (1926): outcomes are random, though probabilities for them can be computed; • Amplitude superposition (1926): there is a linear superposition principle for probability amplitudes; • State superposition (1926): quantum systems can occur in superpositions of measurable states; • Exclusion principle (1926): amplitudes cancel for fermions like electrons to go in the same state; • Interference (1927): probability amplitudes for particles can interfere, potentially destructively; • Uncertainty principle (1927): quantities like position and momenta have related measurement uncertainties; • Hilbert space (1927): states of systems are represented by vectors of amplitudes rather than individual variables; • Field quantization (1927): only discrete numbers of any particular kind of particle can in effect ever exist; • Quantum tunnelling (1928): particles have amplitudes to go where no classical motion would take them; • Virtual particles (1932): particles can occur for short times without their usual energy-momentum relation; • Spinors (1930s): fermions show rotational invariance under SU(2) rather than SO(3); • Entanglement (1935): separated parts of a system often inevitably behave in irreducibly correlated ways; • Quantum logic (1936): relations between events do not follow ordinary laws of logic; • Path integrals (1941): probabilities for behavior are obtained by summing contributions from many paths; • Imaginary time (1947): statistical mechanics is like quantum mechanics in imaginary time; • Vacuum fluctuations (1948): there are continual random field fluctuations even in the vacuum; • Aharonov–Bohm effect (1959): magnetic fields can affect particles even in regions where they have zero strength; • Bell's inequalities (1964): correlations between events can be larger than in any ordinary probabilistic system; • Anomalies (1969): virtual particles can have effects that violate the original symmetries of a system; • Delayed choice experiments (1978): whether particle or wave features are seen can be determined after an event; • Quantum computing (1980s): there is the potential for fundamental parallelism in computations.

In 1876 William Thomson (Kelvin) constructed a so-called harmonic analyzer, in which an assembly of disks were used to sum trigonometric series and thus to predict tides.

As discovered by Jeffrey Shallit in 1979, numbers of the form Sum[1/k 2 i , {i, 0, ∞ }] that have nonzero digits in base k only at positions 2 i turn out to have continued fractions with terms of limited size, and with a nested structure that can be found using a substitution system according to {0, k - 1, k + 2, k, k, k - 2, k, k + 2, k - 2, k} 〚 Nest[Flatten[{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {5, 6}, {3, 4}, {9, 10}, {7, 8}, {9, 10}, {3, 4}} 〚 # 〛 ]&, 1, n] 〛 The continued fractions for square roots are always periodic; for higher roots they never appear to show any significant regularities.

Γ 〚 i 〛 , {i, d}, {j, d}, {k, d}] where the so-called Christoffel symbol Γ ij k is Γ = With[{gi = Inverse[g]}, Table[Sum[ gi 〚 l, k 〛 ( ∂ p 〚 j 〛 g 〚 i, l 〛 + ∂ p 〚 i 〛 g 〚 j, l 〛 - ∂ p 〚 l 〛 g 〚 j, l 〛 ), {l, d}], {i, d}, {j, d}, {k, d}]]/2 There are d 4 elements in the nested lists for Riemann , but symmetries and the so-called Bianchi identity reduce the number of independent components to 1/12 d 2 (d 2 - 1) —or 20 for d = 4 .

(The proof is based on having bounds for how close to zero Sum[ α i , Log[ α i ], i, j] can be for independent algebraic numbers α k .)

In 1941 Richard Feynman pointed out that amplitudes in quantum theory could be worked out by using path integrals that sum with appropriate weights contributions from all possible histories of a system.