SMall Is Beautiful v 0.36

http://sourceforge.net/projects/smib/

philippe.billet@noos.fr

1 Introduction

• number theory (integer, primality, arithmetic functions, Dirichlet [B]  [B] Johann Peter Gustav Lejeune Dirichlet (1805 - 1859) convolution),
• algebra (polynomial, matrix, rational function, tensor),
• analysis (trigonometry, special functions, differential calculus, integration, Fourier transform, Taylor sum),
• differential geometry (curves, surfaces, tensorial calculus),
• numerical analysis (interpolation, numerical integration, ordinary differential equation, sample manipulation, numerical Fourier [C]  [C] Joseph Fourier (1768–1830) transform, differintegration, complex analysis),
• probality & statistic (expected value, variance, standard deviation, skewness, kurtosis, least square fitting, quantile, stochastic differential equation).

2 Compilation & use

↓Compilation

• make
• gcc
• g++

Use

• Interactive mode↓↓: open a terminal, and type ./smib (if you are in source folder, or smib if you copy executable), then you have the prompt messages, it’s up to you.
• ↓↓Script mode: here you must have a valid file (there are some examples in folder /smib/documentation and /smib/documentation/application - in folder /smib/documentation/tutorial gives basics examples). Then type ./smib ./documentation/tutorial, smib treats the file. N.B.: if file ends with quit() then, after treating, smib goes back to shell, else it enters in interactive mode.
• In source directory, the file init.cpp contains many user defined functions, almost all are discribed in this document in section 5 Applications.

Boolean indicators

• If indint=1 then Sint uses Simpson [D]  [D] Thomas Simpson (1710–1761) scheme↓, else Riemann [E]  [E] Georg Friedrich Bernhard Riemann (1826 - 1866) scheme↓.
• If autoexpand=1 then expressions are automatically expanded.
• If cplxmode=1 then complex exponentials are converted into trigonometric form.
• If expomode=1 then ↓trigonometric functions are converted into exponential form.
• If ibpmode=1 then integration by part↓ is used.
• If opemode=1 then operators are automatically expanded.
• prec is iteration number for numerically defined functions.
• precdisc is iteration number for ↓discrete moment.
• precision is used by ↓quantile.
• If simpsonint=1 then numint uses Simpson scheme↓, else Gauss [F]  [F] Johann Carl Friedrich Gauss (1777 - 1855) scheme↓.
  • precintgauss is number of point used for Gauss scheme↓.
  • precintsimpson is number of point used for Simpson scheme↓.

Performance

• analysis
  real 1m1.537s
  user 1m0.752s
  sys 0m0.396s
• geodiff
  real 0m25.007s
  user 0m24.854s
  sys 0m0.060s
• number
  real 21m49.606s
  user 21m43.085s
  sys 0m1.068s
• proba
  real 20m18.279s
  user 20m7.903s
  sys 0m4.592s
• QM
  real 0m0.455s
  user 0m0.388s
  sys 0m0.028s
• tutorial
  real 4m12.296s
  user 4m8.976s
  sys 0m1.936s

3 Mathematical objects

3.1 Numbers

↓Integers

How to define

• ↓ [G]  [G] Marin Mersenne (1588 - 1648)Mersenne number (M_n = 2ⁿ − 1) ↓: ↓mersenneN(n)
• ↓ [H]  [H] Pierre de Fermat (1601 – 1665)Fermat number (F_n = 2²ⁿ − 1)↓: ↓fermatN(n)
• ↓ [I]  [I] Leonardo Fibonacci (1170 – 1250)Fibonacci number (Fib_n = ⎧⎪⎨⎪⎩ 0  n = 0        1  n = 1        Fib_n − 1 + Fib_n − 1  else  )↓: ↓fibonacciN(n)
• ↓ [J]  [J] Eugène Charles Catalan (1814 – 1894)Catalan number (C_n = ^{⎛⎜⎝ 2n   n − 1 ⎞⎟⎠}⁄_n) ↓: ↓catalanN(n)
• ↓↓↓ [K]  [K] James Stirling (1692 – 1770)Stirling numbers↓ of first (S¹_n, m = ∑^{k = n − m}_k = 1( − 1)^j⎛⎜⎝ n − 1 + k   n − m + k ⎞⎟⎠⎛⎜⎝ 2n − k   n − m − k ⎞⎟⎠S²_{n − m + k, k}) and second kind (S²_n, m = ∑^k = m_k = 0( − 1)^{m + k⎛⎜⎝ m   k ⎞⎟⎠kn}⁄_m!): ↓stirlingN1k(n,m), ↓stirlingN2k(n,m)
• ↓↓ [L]  [L] Eric Temple Bell (1883 – 1960)Bell number (B_n = ∑^k = n_k = 1S²_n, k): ↓bellN(n)
• ↓↓ [M]  [M] Angelo Genocchi (1817 – 1889)Genocchi number (^2t⁄_e^t + 1 = ∑^i = ∞_i = 0G_n^tn⁄_n!): ↓genocchiN(n)
• ↓↓↓↓ [N]  [N] Leonhard Euler (1707–1783)Euler numbers of first and second kind (¹⁄_cosh(t) = ∑^i = ∞_i = 0E¹_n^tn⁄_n!; E²_n, k = ∑^j = n_j = 1( − 1)^{j⎛⎜⎝ n + 1   j ⎞⎟⎠}⁄_j!): ↓eulerN1k(n), ↓eulerN2k(n,m)

↓Functions acting on integers

• odd↓ or even↓ - odd: ↓isodd(n); even: ↓iseven(m)
• ↓↓least common multiple & great common divisor↓↓ - gcd: ↓gcd(5,3), lcm: ↓lcm(5,3)
• counting: factorial (n!=∏i = ni = 1i)↓: 2!↓, binomial coefficient (⎛⎜⎝ n   k ⎞⎟⎠ = n!⁄k!(n − k)!)↓: ↓binomial(5,3), ↓permutation (per(n, k) = ⎛⎜⎝ n   k ⎞⎟⎠k!): ↓permutation(5,3), ↓derangement (der(n) = !n = n!∑i = ni = 0( − 1)j⁄j!): ↓derangement(5)
• integer factorization & primality test: n = ∏i = ki = 1pαii is the factorization in prime factors↓ of n: ↓factor(n), ↓primedecomp(n) returns 2 vectors the first contains prime factors, the second exponents of corresponding factor; primality test↓: ↓isprime(n)
• number theory functions & operators:
  • operators acting on arithmetic function:
    sum over divisors↓↓ (∑d|Nf(d)): ↓sumoverdivisor(x,N,f),
    sum over primes↓↓ (∑p|Nf(p)): ↓sumoverprime(x,N,f),
    product over divisors↓↓ (∏d|Nf(d)): ↓productoverdivisor(x,N,f),
    product over primes↓↓ (∏p|Nf(p)): ↓productoverprime(x,N,f),
    Dirichlet convolution product↓↓ (f⋆g(n) = ∑d|nf(d)g(n⁄d)): ↓dirichletproduct(f,g,x,n)
  • functions (we assume that :n = ∏i = ki = 1pαii):
    number of distinct prime factors (ω(n) = k)↓:↓omega(n)
    total number of prime factors (Ω(n) = ∑i = ki = 1αi)↓: ↓Omega(n)
    number of divisors (τ(n) = 1⋆1(n))↓ ↓: ↓tau(n)
    prime counting (π(n) = ∑i = ni = 21P(i), P is the set of prime numbers)↓: ↓piprime(n)
    sum of divisors at ordre r (σr(n) = ∑d|ndr)↓: ↓sigma(n,r)
    kernel of integer (ker(n) = ∏i = ki = 1pi)↓ ↓: ↓integerkernel(n)
    ↓ [O]  [O] Joseph Liouville (1809 – 1882)Liouville function (L(n) = ( − 1)Ω(n))↓: ↓liouvillefunction(n)
    ↓Euler totient function (φ(n) = n∏i = ki = 1(1 − 1⁄pi))↓: ↓eulerphi(n)
    ↓↓ [P]  [P] August Ferdinand Möbius (1790 – 1868)Möbius function (μ(n) = ⎧⎨⎩ ( − 1)ω(n)  ω(n) = Ω(n)        0  ω(n) < Ω(n)  )↓: mobius(n)
    ↓ [Q]  [Q] Hans Carl Friedrich von Mangoldt (1854–1925)von Mangoldt function (Λ(n) = ⎧⎨⎩ ln(p)  n = pk,  p ∈ P        0  otherwise  )↓: ↓mangoldt(n)
     [R]  [R] Adrien-Marie Legendre (1752–1833)Legendre symbol (⎛⎝( a )/(n)⎞⎠L = an − 1⁄2mod(n))↓↓: ↓legendresymbol(n,m)
     [S]  [S] Carl Gustav Jacob Jacobi (1804–1851) Jacobi symbol (⎛⎝( a )/(n)⎞⎠J = ∏i = ki = 1⎛⎝( a )/(pi)⎞⎠αiL)↓↓: ↓jacobisymbol(n,m)
    ↓↓ [T]  [T] Franz Mertens (1840–1927)Mertens function (M(n) = ∑i = ni = 1μ(k)): ↓mertens(n)

↓Rational numbers

How to define

• ↓ [U]  [U] Jacob Bernoulli (also known as James or Jacques) (1654–1705)Bernoulli number (^t⁄_{e^t − 1} = ∑^i = ∞_i = 0B_n^tn⁄_n!)↓: ↓bernoulliN(n)

Functions acting on rational numbers

• ↓numerator(q) returns numerator↓ of a rational
• ↓denominator(q) returns denominator↓ of a rational
• ↓rationalize(q) ↓reduces of rational expression

↓Real numbers

How to define

• ↓ [V]  [V] Archimedes of Syracuse (287 BC–212 BC)Archimedes constant π↓: pi↓,
• ↓unity exponential e: exp(1),
• ↓↓Euler constant γ: ↓euler.

Functions acting on real numbers

Complex numbers↓

How to define

Functions acting on complex numbers

• ↓real(z) returns real part↓ of z (i.e. ℜ(z) = a)
• ↓imag(z) returns ↓imaginary part of z (i.e. ℑ(z) = b)
• ↓abs(z) returns absolute value↓ of z (i.e. |z| = √(a² + b²))
• ↓arg(z) returns ↓argument of z (i.e. arg(z) = arctan(^ℑ(z)⁄_ℜ(z)))
• ↓conj(z) returns conjugate↓ of z (i.e. z = a − ib)

Functions acting on any type of numbers

• Basic operations: +, -, *, /, ^↓ ↓↓↓↓
• Exponential↓ & ↓logarithmic function↓: ↓exp(z), ↓log(z)
• ↓↓Trigonometric functions: sin(z), cos(z), tan(z), sec(z), csc(z), cot(z)↓↓↓↓↓↓
• ↓↓Inverse trigonometric functions: arcsin(z), arccos(z), arctan(z)↓↓↓
• ↓Hyperbolic trigonometric functions↓: sh(z), ch(z), th(z), sech(z), csch(z), coth(z)↓↓↓↓↓↓
• ↓Inverse hyperbolic trigonometric functions↓: argsh(z), argch(z), argth(z) ↓↓↓
• ↓ [Y]  [Y] Christoph Gudermann (1798-1852)Guderman function↓: gd(z), arggd(z)↓↓
• Γ and B functions: ↓ ↓Gamma(z), Beta↓↓(z1,z2)↓ ↓
• ↓Error functions↓: erf(z), erfc(z)↓
• ↓ [Z]  [Z] Friedrich Wilhelm Bessel (1784-1846)Bessel functions↓: besseli(z,n), besselj(z,n), besselk(z,n) ↓.

Variables

How to define

Functions acting on variables

3.2 Structures of higher dimensions

↓Vectors

How to define

Functions acting on vectors

↓Matrices

How to define

Functions acting on matrices

• Transposition↓: ↓trans(M,i,j) where i, j are indices to transpose (for matrices i = 1 and j = 2,  for tensors it is a little bit different).
• Determinant↓ of a matrix: ↓det(M).
• Inverse↓ of a matrix: ↓inv(M).
• Trace ↓of a matrix: ↓trace(M).
• ↓ [A]  [A] Ferdinand Georg Frobenius (1849-1917)Frobenius norm (∥M∥ = √(M.M^t)): ↓frobeniusnorm(M).
• ↓Product of matrix by vector: ↓dot(M,X).
• Characteristic polynomial (char(M, x) = det(M − xI))↓↓: ↓charpoly(M), here the polynomial variable is x.
• Eigenvalues↓ and ↓eigenvectors of symmetric matrix M : eigen(M)↓, eigenvalues are stored in diagonal matrix D and eigenvectors are stored in matrix Q, and we have got :M = Q^TDQ.

↓Polynomials

How to define

• ↓Cyclotomic polynomial (Φ_n(x) = ∏_d|n(1 − x^n⁄_d)^μ(d))↓: ↓cyclotomic(x,n)
• ↓Orthogonal polynomials↓:
  • ↓ [B]  [B] Charles Hermite (1822-1901)Hermite polynomials (H_n(x) = ⎧⎪⎨⎪⎩ 1  n = 0        2x  n = 1        2xH_n − 1(x) − 2(n − 1)H_n − 2(x)  otherwise  )↓: ↓hermite(x,n)
  • ↓ [C]  [C] Edmond Nicolas Laguerre (1834-1886) Laguerre polynomials
    (L_n, k(x) = ⎧⎪⎨⎪⎩ 1  n = 0         − x + k + 1  n = 1        ¹⁄_n((2(n − 1) + 1 − x + k)L_{n − 1, k}(x) − (n − 1 + k)L_{n − 2, k}(x))  otherwise  )↓: ↓laguerre(x,n,k)
  • ↓↓ [D]  [D] Adrien-Marie Legendre (1752-1833)Legendre polynomials
    (P_n(x) = ⎧⎪⎨⎪⎩ 1  n = 0        x  n = 1        (1)/(n)((2(n − 1) + 1)xP_n(x) − (n − 1)P_n − 1(x))  otherwise  ): ↓legendre(x,n,m)
  • U & T ↓ [E]  [E] Pafnouti Lvovitch Tchebychev (1821-1894)Tchebychev polynomials↓(T_n(x) = ⎧⎪⎨⎪⎩ 1  n = 0        x  n = 1        2xT_n − 1(x) − T_n − 2(x)  otherwise  ; U_n(x) = ⎧⎪⎨⎪⎩ 1  n = 0        2x  n = 1        2xU_n − 1(x) − U_n − 2(x)  otherwise  ):
    ↓tchebychevU(x,n) & ↓tchebychevT(x,n).

Functions acting on polynomials

• Basic operations: +, -, *, ^ (exponent must be an integer)
• Degree of a polynomial↓ relatively to a variable: ↓deg(P(x),x)
• Divisibility:
  • ↓ [F]  [F] Euclid of Alexandria (????-????)Euclidian division (returns Q and R such that A = BQ + R, deg(R) < deg(B) : ↓divpoly(A(x),B(x),x)
  • Greater common divisor↓ of polynomials: gcdpoly(P(x),Q(x),x)↓
  • ↓Extented greater common divisor of polynomials (↓↓ [G]  [G] Étienne Bézout (1730–1783)Bezout identity i.e. returns A and B such that PA + QB = R) : egcdpoly(P(x),Q(x),R(x),x)↓
• Square free factorization↓ of a polynomial : squarefreepoly(P(x),x)↓ using ↓ [H]  [H] David Yun (????-????)Yun square free factorization.
• ↓ Resultant of polynomials : resultantpoly(P(x),Q(x),x)↓
• ↓ Discriminant of a polynomial : discpoly(P(x),x)↓
• ↓Increasing power division: ↓ incrdivpoly(P(x),Q(x),x,n)

↓Rational function

How to define

Functions acting on samples

↓Samples

How to define

Functions acting on samples

• ↓Snorm↓(S)↓: computes:
  (√(∑_{i ∈ {0..n}}∥S_i, 2∥²), ^{∑_{i ∈ {0..n}}∥S_i, 2∥}⁄_n, √(∑_{i ∈ {0..n}}{∥S_i, 2∥ − ^{∑_{i ∈ {0..n}}∥S_i, 2∥}⁄_n}²))
• ↓Ssum(S,T): sum of two samples↓↓ {(x_i, S_i, 2 + T_i, 2)}_{i ∈ {0..n}}
• ↓Sdiff(S,T): difference between two samples↓↓{(x_i, S_i, 2 − T_i, 2)}_{i ∈ {0..n}}
• ↓Sproduct(S,T): product of two samples↓↓{(x_i, S_i, 2.T_i, 2)}_{i ∈ {0..n}}
• ↓Sscalarproduct(z,S): ↓↓product of a scalar and a sample {(x_i, z.S_i, 2)}_{i ∈ {0..n}}
• ↓Sreal(S): ↓real part of a sample↓ {(x_i, ℜ(S_i, 2))}_{i ∈ {0..n}}
• ↓Simag(S): ↓imaginary part of a sample↓{(x_i, ℑ(S_i, 2))}_{i ∈ {0..n}}
• ↓Sabs(S): absolute value of a sample↓↓ {(x_i, ∣S_i, 2∣)}_{i ∈ {0..n}}
• ↓Sarg(S): ↓↓argument of a sample {(x_i, arg(S_i, 2))}_{i ∈ {0..n}}
• ↓Sd(S): ↓↓derivative of a sample {(x_i, ^{S_i + 1 − S_i}⁄_{xi + 1 − xi})}_{i ∈ {0..n}}(in fact, it is a little bit more complicated because there are some problems at boundary).
• ↓Sfracd(S,q,x0): ↓↓ fractional derivative of a sample to a non integer order q at a point x₀ (cf. § 5.5.8).
• ↓Sint(S): ↓↓integral of a sample (if indint equals 0 then Sint uses ↓↓Riemann method, if indint equals 1 then Sint uses  [I]  [I] Thomas Simpson (1710-1761)Simpson method↓↓).
• ↓Sdefint(S,a,b): ↓↓ definite integral of a sample (if S is defined on [c, d], integral is computed on [a, b]∩[c, d]).
• ↓Sample filtering:
  • Sfilterxinf(S,a): cancellation of all values less than a
  • Sfilterxsup(S,a): cancellation of all values greater than a
  • Sfilterxs(S,a,b): cancellation of all values outside [a, b]
• ↓Sample truncation:
  • Struncatexinf(S,a): truncation of all values less than a
  • Struncatexsup(S,a): truncation of all values greater than a
  • Struncatexs(S,a,b): truncation of all values outside [a, b]

• ↓Sdiscfourier(S,a,b): Fourier tranform of sample↓↓, here sample must be defined on a symetric interval [ − a, a] the result is defined on [ − b, b], for inverse Fourier transform: ↓Sdiscinvfourier(S,b,a). ↓Sdiscconvolution(S,T,a,b): convolution product of two samples↓ (they must be defined on [ − a, a], and result is defined on [ − b, b].
• Sreal2, Simag2, Sabs2, Sarg2, Sproduct2, Ssum2, Sscalarproduct2 are functions for samples defined like this {(x_i, y_i, f(x_i, y_i))}_{i ∈ {0..n}}.
• With Sdecomp↓ and ↓Srecomp, we can respectively extract a column of a sample, and glue columns (only for 2D samples). With Sdelete↓, we can delete a column of any sample (cf.  [J]  [J] Edward Norton Lorenz (1917–2008)Lorenz↓ or  [K]  [K] Otto Eberhard Rössler (1940-????) Rössler system↓).
• parametricsample produces a 3D-sample, let (u, v) ∈ [a₁, b₁] × [a₂, b₂]⟼(v₁(u, v), v₂(u, v), v₃(u, v)), parametricsample((v1(u,v),v2(u,v),v3(u,v)),u,v,-a,-b,n) computes sample{(v₁(u, v), v₂(u, v), v₃(u, v))}_{(u, v) ∈ [ − a, a] × [ − b, b]}

↓Tensors

• ↓↓ [L]  [L] Elwin Bruno Christoffel (1829-1900)Christoffel symbol (or ↓connection coefficient): Γ^α_μν = ¹⁄₂g^αβ(g_βμ, ν + g_βν, μ − g_μν, β),
• Riemann tensor↓↓: R^α_βγδ = Γ^α_βδ, γ − Γ^α_βγ, δ + Γ^α_μγΓ^μ_βδ − Γ^α_μδΓ^μ_βγ,
• the  [M]  [M] Gregorio Ricci-Curbastro (1853-1925)Ricci tensor↓ ↓: R_μν = R^α_μαν,
• the ↓↓↓Ricci scalar (or Ricci curvature): R = R_μμ,
• the ↓↓ [N]  [N] Albert Einstein (1879-1955)Einstein tensor: G_μν = R_μν − ¹⁄₂g_μνR.

• ↓↓covariant derivative: T^α_;γ = T^α_, γ − T^μΓ^α_μγ
• ↓commutator: [u, v] = (u^βv^α_, β − v^βu^α_, β)e_α
• ↓↓directed covariant derivative: ∇_uv = v^α_;βu_α(using that we have too:[u, v] = ∇_vu − ∇_uv)
• ↓↓covariant derivatives of tensor:
  • T^αβ_;γ = T^αβ_, γ + T^μβΓ^α_μγ + T^αμΓ^β_μγ
  • T^α_β;γ = T^α_β, γ + T^μ_βΓ^α_μγ − T^α_μΓ^μ_βγ
  • T_αβ;γ = T_αβ, γ − T_μβΓ^μ_αγ − T_αμΓ^μ_βγ

3.3 Operators & functions

Functions

• ↓sum(n,n0,n1,f(n)): partial sum↓↓ (∑^n = n1_n = n0f(n))
• ↓product(n,n0,n1,f(n)): partial product↓↓ (∏^n = n1_n = n0f(n))
• ↓nroot(P,x): numerical computation of ↓roots of a polynomial↓ using ↓↓ [P]  [P] Isaac Newton (1642–1727)Newton method.
• ↓simpson(f(x),x,a,b,n): Simpson numerical scheme↓↓ of integration
• ↓numint(f(x),x,a,b,n): numerical scheme of integration↓, if simpsonint=1, numint uses Simpson scheme↓, else Gauss scheme↓.

Operators

• ↓simplify(exp): ↓simplification of expression
• ↓expand(exp): ↓expansion of expression
• ↓limit(f(x),x,x0): ↓limit of f relatively to x at x₀
• ↓d(f(x),x,n): ↓derivative of f relatively to x to order n
• ↓odesolve(F(d(f(x),x),f(x)),f(x),x): first and second order linear unhomogeneous ordinary differential equation solver↓↓
• ↓taylor(f(x),x,x0,n):  [Q]  [Q] Brook Taylor (1685-1731)Taylor partial sum↓ ↓↓ of f relatively to x at x₀ to order n
• ↓integral(f(x),x,n): integral↓ of f relatively to x to order n
• ↓antider(f(x),x): antiderivative↓ of f relatively to x (search F such that F’ = f).
• ↓ibp(f(x),g(x)): integral by parts↓ of f and g
• ↓fourier(f(x),x) & ↓invfourier(f(x),x): Fourier transform↓ & inverse (ℱ(f)(t) = ∫_ℝf(x)e^− itxdt & ℱ^− 1(f)(t) = ∫_ℝf(x)e^itxdx ⁄ (2π))
• ↓defint(f(x),x,a,b): integral↓ of f relatively to x on [a, b] (this function uses antider, integral, fourier (using ∫_ℝf.dμ = f̂(0))).
• ↓convolution(f(x),g(x)): ↓convolution product
• ↓translation↓(f(x),x,a): f(x − a)
• ↓dilatation↓(f(x),x,a): f(ax)
• ↓modulation↓(f(x),x,a): e^iaxf(x)

  
  

4 Elements of language

↓prog

↓Condition

• ↓equality: ==
• ↓inequality: < or >
• ↓isinteger
• ↓logic disjonction: or
• ↓logic conjonction: and
• ↓logic negation: not

↓Loop

• ↓for(variable,initial value, final value,action to do), for is very practice for every indiced objects (vectors, matrices,...)
• recursive loop↓: an example is worth a thousand words gcd of two polynomials:
  
  gcdpoly(a,b,x)=prog(temp1,
  do(test(degree(a)<degree(b),
  gcdpoly(b,a,x),
  do(test(b==0,a,gcdpoly(b,divpoly(a,b,x)[2],x))))))

↓nil↓

Dynamic allocation of array : ↓Syracuse problem

↓Plotting

5 Applications

• arithmetic & number theory:
  • primality tests
  • arithmetic functions properties
  • Dirichlet’s product properties
  • π computation
  • complex number and quaternion
• function manipulation applied to quantum mechanic:
  • operator (creation, annihilation, commutator...)
  • operator acting on function
  • relation between quantum mechanic and Hermite function
• differential geometry:
  • theory of curves: planar & 3D
  • gaussian theory of surfaces
  • riemannian geometry
• unidimensional calculus:
  • classical integrals
  • orthogonal polynomial properties
  • complex path integrals
  • continous Fourier transform:
    • elementary functions
    • Fourier transform & convolution product
    • Fourier transform & Green functions
• vectorial calculus:
  • vectorial integrals
  • curvilinear integrals
  • area integrals
  •  [R]  [R] George Green (1793-1841)Green theorem
  •  [S]  [S] George Gabriel Stokes (1819-1903)Stokes theorem
  • Gauss-Ostrogradski [T]  [T] Mikhail Vasilyevich Ostrogradsky (1801-1862) theorem
• numerical analysis:
  • univariate calculus applied to samples
  • special function (Bessel,  [U]  [U] Hermann Hankel (1839-1873)Hankel,  [V]  [V] George Biddell Airy (1801–1892)Airy...)
  • interpolation ( [W]  [W] Joseph-Louis Lagrange (1736-1813)Lagrange-Newton, least squares mean)
  • polynomial interpolation
  • numerical integration
  • discrete Fourier transform using samples:
    • elementary functions
    • differential equations
    • pseudo-differential operator
    • wavelets
  • complex analysis
  • ordinary differential equation
  • fractional calculus
• probability & statistic:
  • distributions
  • moments
  • quantile & median
  • random numbers generation
  • law of large numbers & central limit theorem
  • stochastic differential equation

5.1 Arithmetic, Number theory & Polynomial

5.1.1 Computing famous numbers

Using direct computation

Using recursive computation

Using generating function

5.1.2 ↓primality tests

↓Deterministic test: using ↓ [X]  [X] John Wilson (1741-1793)Wilson formula↓

↓Probabilistic test: ↓ [Y]  [Y] Robert Martin Solovay (1938-????)Solovay- [Z]  [Z] Volker Strassen (1936-????)Strassen test

5.1.3 properties of arithmetic functions↓↓

↓Sum over divisors

• ∑_d|N|μ(d)|φ(d) = γ(N)
• ∑_d|N¹⁄_d² = ^σ(N, 2)⁄_N²
• ∑_d|N|μ(d)| = 2^ω(N)
• ∑_d|Nμ(d)τ(d) = ( − 1)^ω(N)
• ∑_d|Nμ(d)λ(d) = 2^ω(N)
• ∑_d|Nσ(d, 1) = N.∑_d|N^τ(d)⁄_d

Product over divisors↓ (or prime divisors↓)

• ∏_d|Nd = N.^τ(N)⁄₂
• ( − 1)^ω(N)∏_p|Np = ∑_d|Nμ(d)σ(d, 1)
• ( − 1)^ω(N)∏_p|N(p − 2) = ∑_d|Nμ(d)φ(d).

↓↓Dirichlet product

• σ = id⋆1
• τ = 1⋆1
• φ = μ⋆id
• σ = φ⋆τ
• id = σ⋆μ
• id = φ⋆1
• id.τ = φ⋆σ
• (id.σ)(n) = ∑_d|nτ(d)
• (id².σ)(n) = ∑_d|nd.σ(d)
• σ⋆σ = (id.τ)⋆τ
• ∑_1≤n≤N(τ⋆μ⋆Λ)(n) = log(N!)(here X = (1, 2, 3, 4, 20, 25, 154))
  

5.1.4 ↓↓Arithmetic equations

• ↓ [A]  [A] Enrico Bombieri (1940-????)Bombieri conjecture

• 4τ(n + 2) = φ(n),  0≤n≤1000

• σ(n, 1) = 2n + 1,  0≤n≤1000

5.1.5 Perfect number↓ & harmonic mean of divisors↓

5.1.6 Mertens function & Redheffer matrix

5.1.7 Arithmetic & statistic

• statkurtosis(listprime(n)) and ku(n), and smib gives:

• statskewness(listprime(n)) and sk(n), and smib gives:

5.1.8 Polynomial↓

5.1.9 Sum↓ &  [C]  [C] Bill Gosper (1943-????) Gosper algorithm↓↓

5.2 Unidimensional integral↓ calculus↓

5.2.1 ↓Antiderivative

• In the directory /smib/documentation/application/Performance, we find the file testintegral which contains about 300 integrals, at the moment smib gives a good answer at 92% at least.
• we are very far from  [D]  [D] Robert Henry Risch (1939-????)Risch algorithm↓.

5.2.2 Classical integrals

• ∫_{[ − ∞, ∞]}1_[a, b]dx
• ∬_{[a1, b1] × [a2, b2]}1dx
• ∫_{[ − ∞, ∞]}e^− x2dx
• ∫_{[ − ∞, ∞]}¹⁄_{(1 + x²)}dx
• ∫_[0, ∞]^sin(x)⁄_xdx
• ∫_{[ − ∞, ∞]}^sin(x)⁄_xdx
• ∫_[0, ∞]e^− xdx.

5.2.3 ↓↓Orthogonal polynomial properties

5.2.4 ↓↓Complex path integrals

5.2.5 ↓Continous Fourier transform↓

• On functions:
  a constant: print("fourier(a,t)=",fourier(a,t))
  sin(t): print("fourier(sin(t),t)=",expand(fourier(sin(t),t)))
  e^− a|t|: print("fourier(exp(-a*abs(t)),t)=",fourier(exp(-a*abs(t)),t))
  log(a|t|): print("fourier(log(a*abs(t)),t)=",fourier(log(a*abs(t)),t))
  e^− t2: print("fourier(exp(-t^2),t)=",fourier(exp(-t^2),t))
  e^it2: print("fourier(exp(i*t^2),t)=",fourier(exp(i*t^2),t))
  arctan(t): print("fourier(arctan(t),t)=",fourier(arctan(t),t))
  erf(t): print("fourier(erf(t),t)=",fourier(erf(t),t))
  
• On operators:
  ∂²_xf: print("fourier(d(f(t),t,2),t)=",fourier(d(f(t),t,2),t))
  f⋆g: print("fourier(convolution(f(t),g(t)),t)=",
              expand(fourier(convolution(f(t),g(t)),t)))
  ∫f: print("fourier(integral(f(t),t),t)=",fourier(integral(f(t),t),t))
• Applications:
  ℱ^− 1(¹⁄_{ℱ(^dδ(t)⁄dt − δ(t))}):
  print("invfourier(1/fourier(d(dirac(t),t)-dirac(t),t),t)=",
  invfourier(1/fourier(d(dirac(t),t)-dirac(t),t),t))
  print("d(",last,",t)-",last,"=",d(last,t)-last)
  ℱ^− 1(¹⁄_{ℱ(t^dδ(t)⁄dt − δ(t))}):
  print("invfourier(1/fourier(t*d(dirac(t),t)-dirac(t),t),t)=",
  invfourier(1/fourier(t*d(dirac(t),t)-dirac(t),t),t))
  print("t*d(",last,",t)-",last,"=", t*d(last,t)-last)
  t∂_tδ (it is just an application of the following propertie (f.g = ℱ^− 1(ℱ(f)⋆F(g)):
  print("fourierprod(d(dirac(t),t),t)=", expand(fourierprod(d(dirac(t),t),t)))

• f⋆sgn = 2∫f:
  print("convolution(f(x),sgn(x))=", expand(convolution(f(x),sgn(x))))
• ¹⁄_x⋆¹⁄_x =  − π²δ(x):
  print("convolution(1/x,1/x)=", expand(convolution(1/x,1/x)))
• ¹⁄_x²⋆¹⁄_x² =  − π^2dδ(x)⁄_dx:
  print("convolution(1/x^2,1/x^2)=", expand(convolution(1/x^2,1/x^2)))
• e^− t2⋆e^− t2 = √(^π⁄₂)e^{− t2⁄₂}:
  print("convolution(exp(-t^2),exp(-t^2))=",
  expand(convolution(exp(-t^2),exp(-t^2))))

• (H): ∂_tf − ∂²_xf:
  d(f(x,t),t)-d(f(x,t),x,2)
  fourier(last,x)
  expand(last)
  odesolve(last,fourier(f(x,t),x),t)
  invfourier(last,x)
  d(last,t)-d(last,x,2) (I)
  
  We have the following symbolic solution:
  
  If we sample (I), we can see that it is all zero: symbolic solution is correct.
• (S): ∂_tf − i∂²_xf:
  d(f(x,t),t)-i*d(f(x,t),x,2)
  fourier(last,x)
  expand(last)
  odesolve(last,fourier(f(x,t),x),t)
  invfourier(last,x)
  d(last,t)-i*d(last,x,2) (II)
  
  We have the following symbolic solution:
  
  For (II) as for (I), the sample is all zero: symbolic solution is correct.

5.3 ↓Vectorial calculus

5.3.1 ↓↓Differential operators in↓ ↓orthogonal coordinate

• gradient of a scalar field ∇f :
  grado(f,Transf,Coord)=prog(FF,H,gradtemp,do(
  FF=firstform(Transf,Coord),
  H=(sqrt(FF[1,1]),sqrt(FF[2,2]),sqrt(FF[3,3])),
  gradtemp=(1/H[1]*d(f,Coord[1]),1/H[2]*d(f,Coord[2]),1/H[3]*d(f,Coord[3])),
  gradtemp))
• divergence of a vector field ∇.V :
  divo(V,Transf,Coord)=prog(FF,H,divtemp,do(
  FF=firstform(Transf,Coord),
  H=(sqrt(FF[1,1]),sqrt(FF[2,2]),sqrt(FF[3,3])),
  divtemp=1/(H[1]*H[2]*H[3])*(d(V[1]*H[2]*H[3],Coord[1])+
  d(V[2]*H[3]*H[1],Coord[2])+
  d(V[3]*H[1]*H[2],Coord[3])),
  divtemp))
• curl of a vector field ∇∧V :
  curlo(V,Transf,Coord)=prog(FF,H,curltemp,do(
  FF=firstform(Transf,Coord),
  H=(sqrt(FF[1,1]),sqrt(FF[2,2]),sqrt(FF[3,3])),
  curltemp=(1/(H[2]*H[3])*(d(H[3]*V[3],Coord[2])-d(H[2]*V[2],Coord[3])),
  1/(H[1]*H[3])*(d(H[1]*V[1],Coord[3])-d(H[3]*V[3],Coord[1])),
  1/(H[1]*H[2])*(d(H[2]*V[2],Coord[1])-d(H[1]*V[1],Coord[2]))),
  curltemp))
• laplacian of a scalar field Δf :
  laplo(f,Transf,Coord)=prog(FF,H,lapltemp,do(
  FF=firstform(Transf,Coord),
  H=(sqrt(FF[1,1]),sqrt(FF[2,2]),sqrt(FF[3,3])),
  lapltemp=1/(H[1]*H[2]*H[3])*(d(H[2]*H[3]/H[1]*d(f,Coord[1]),Coord[1])+
  d(H[3]*H[1]/H[2]*d(f,Coord[2]),Coord[2])+
  d(H[1]*H[2]/H[3]*d(f,Coord[3]),Coord[3])),
  lapltemp))
• laplacian of a vector field ΔV :
  vlaplo(V,Transf,Coord)=grado(divo(V,Transf,Coord),Transf,Coord)-
  curlo(curlo(V,Transf,Coord),Transf,Coord)

5.3.2 ↓↓Vectorial integrals

5.3.3 ↓↓Curvilinear integrals

• Let C = {x(t) = 2t, t ∈ [0, 1]} a planar curve, andy(t) = 2t a function, we want to compute ∫_Cy²ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=2t
  y=2t
  print("defint(y^2*d(x,t),t,0,1)=",defint(y^2*d(x,t),t,0,1))
  
  And smib gives:
  
  defint(y^2*d(x,t),t,0,1)= 8/3
  
• Let C = {(x(t) = 2t + 1, y(t) = t², z(t) = 1 + t³), t ∈ [0, 1]} a curve in ℝ³, and A⃗ = (z, x, y) a vector, we want to compute ∫_CA⃗.ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=2t+1
  y=t^2
  z=1+t^3
  f=z*d(x)+x*d(y)+y*d(z)
  print("defint(z*d(x)+x*d(y)+y*d(z),t,0,1)=",defint(f,t,0,1))
  
  And smib gives:
  
  defint(z*d(x)+x*d(y)+y*d(z),t,0,1)= 163/30
  
• Let C = {(x(t) = t² + 1, y(t) = 2t², z(t) = t³), t ∈ [1, 2]} a curve in ℝ³, and a vector F⃗ = (3xy,  − 5z, 10x), we want to compute ∫_CF⃗.ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  F=quote(F)
  G=quote(G)
  C=quote(C)
  x=t^2+1
  y=2*t^2
  z=t^3
  F=(3*x*y,-5*z,10*x)
  C=(x,y,z)
  print("defint(dotP(F,d(C,t)),t,1,2)=",defint(dotP(F,d(C,t)),t,1,2))
  
  And smib gives:
  
  defint(dotP(F,d(C,t)),t,1,2)= 303
  
• Let C = {(x(t) = t², y(t) = 2t, z(t) = t³), t ∈ [1, 2]} a curve in ℝ³,φ = 2xyz² a function and F⃗ = (xy,  − z, x²) a vector, we want to compute ∫_Cφds and ∫_CF⃗∧ds. So in smib:
  
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  F=quote(F)
  phi=quote(phi)
  G=quote(G)
  C=quote(C)
  x=t^2
  y=2*t
  z=t^3
  phi=2*x*y*z^2
  F=(x*y,-z,x^2)
  C=(x,y,z)
  print("defint(phi*d(C,t),t,0,1)=",defint(phi*d(C,t),t,0,1))
  print("defint(crossP(F,d(C,t)),t,0,1)=",defint(crossP(F,d(C,t)),t,0,1))
  
  And smib gives:
  
  defint(phi*d(C,t),t,0,1)= (8/11,4/5,1)
  defint(crossP(F,d(C,t)),t,0,1)= (-9/10,-2/3,7/5)

5.3.4 Area integrals↓↓

• Let S = {(x, y, x² + 2y), (x, y) ∈ [0, 1]²} a map in ℝ³, we want to compute ∬_SdS. So in smib:
  
  a=quote(a)
  t=quote(t)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  z=x^2+2y
  S=(x,y,z)
  a=abs(crossP(d(S,x),d(S,y)))
  print("S=",S)
  print("defint(defint(abs(crossP(d(S,x),d(S,y))),x,0,1),y,0,1)=",
  defint(defint(a,x,0,1),y,0,1))
  
  And smib gives:
  
  S= (x,y,2*y+x^2)
  
  defint(defint(abs(crossP(d(S,x),d(S,y))),x,0,1),y,0,1)=
  1/2*(3-5/2*log(2)-5/4*log(5)+5/2*log(10))
  
• Let S = {(ucos(v), usin(v), v), (u, v) ∈ [0, 1] × [0, 3π]} a map in ℝ³, we want to compute ∬_SdS. So in smib:
  
  a=quote(a)
  t=quote(t)
  u=quote(u)
  v=quote(v)
  x=quote(x)
  y=quote(y)
  z=quote(z)
  x=u*cos(v)
  y=u*sin(v)
  z=v
  S=(x,y,z)
  a=abs(crossP(d(S,u),d(S,v)))
  a=simplify(a)
  print("S=",S)
  print("defint(defint(abs(crossP(d(S,u),d(S,v))),u,0,1),v,0,3pi)=",
  defint(defint(a,u,0,1),v,0,3pi))
  
  And smib gives:
  
  S= (u*cos(v),u*sin(v),v)
  
  defint(defint(abs(crossP(d(S,u),d(S,v))),u,0,1),v,0,3pi)=
  -3/2*pi*log(2)+3/2*pi*log(2+2*2^(1/2))+3*pi/(2^(1/2))

5.3.5 Some famous theorems

1. Let S a map in ℝ³defined by S = {(x, y, 4 − x² − y²), (x, y) ∈ [ − 1, 1]²}, and ⟶F = (y, z, x)
2. Let S a map in ℝ³ defined by S = {(x, y, √(1 − x² − y²)), (x, y) ∈ [ − 1, 1]²}, and ⟶F = (2x − y,  − x²y,  − y²z).

5.4 Differential geometry↓

5.4.1 Curves

↓↓Planar curves

↓3D curves

5.4.2 ↓Theory of surfaces

• ↓↓first form, and its determinant↓,
• ↓↓normal vector,
• ↓↓second form,
• ↓↓ [F]  [F] Weingarten (????-????)Weingarten operator,
• ↓↓Mean curvature, Gaussian curvature↓↓, and principal curvature↓↓,
• ↓Shape index↓ and curvedness↓.

5.4.3 ↓Riemannian geometry

• ↓↓Nil: ds² = dx² + dy² + ( − xdy + dz)²
  X=(x,y,z) print("Nil")
  A=((1,0,0),(0,1+x^2,-x),(0,-x,1))
  riemann(A,X)
  print("Metric: gdd=",gdd)
  print("Connection: GAMUDD=",GAMUDD)
  print("Riemann tensor RUDDD=",RUDDD)
  print("Ricci tensor RDD=",RDD)
  print("Ricci Curvature R=",R)
  f=quote(f)
  print("LaplF(f(X),X)=",LaplF(f(X),X))
  print(" ")
  
  And smib gives:
  Metric: gdd= ((1,0,0),(0,1+x^2,-x),(0,-x,1))
  Connection: GAMUDD=
  (((0,0,0),(0,-x,1/2),(0,1/2,0)),
  ((0,1/2*x,-1/2),(1/2*x,0,0),(-1/2,0,0)),
  ((0,-1/2+1/2*x^2,-1/2*x),(-1/2+1/2*x^2,0,0),(-1/2*x,0,0)))
  Riemann tensor RUDDD=
  ((((0,0,0),(0,0,0),(0,0,0)),
  ((0,-3/4+1/4*x^2,-1/4*x),(3/4-1/4*x^2,0,0),(1/4*x,0,0)),
  ((0,-1/4*x,1/4),(1/4*x,0,0),(-1/4,0,0))),
  (((0,3/4,0),(-3/4,0,0),(0,0,0)),
  ((0,0,0),(0,0,-1/4*x),(0,1/4*x,0)),
  ((0,0,0),(0,0,1/4),(0,-1/4,0))),
  (((0,x,-1/4),(-x,0,0),(1/4,0,0)),
  ((0,0,0),(0,0,-1/4-1/4*x^2),(0,1/4+1/4*x^2,0)),
  ((0,0,0),(0,0,1/4*x),(0,-1/4*x,0))))
  Ricci tensor RDD=
  ((-1/2,0,0),(0,-1/2+1/2*x^2,-1/2*x),(0,-1/2*x,1/2))
  Ricci Curvature R= -1/2
  LaplF(f(X),X)=
  d(d(f((x,y,z)),x),x)+d(d(f((x,y,z)),y),y)+d(d(f((x,y,z)),z),z)+
  2*x*d(d(f((x,y,z)),y),z)+x^2*d(d(f((x,y,z)),z),z)
  
• Sol↓↓: ds² = e^2xds² + e^− 2xdy² + dz²
  X=(x,y,z)
  print("Sol")
  A=((exp(2*x),0,0),(0,exp(-2*x),0),(0,0,1))
  riemann(A,X)
  print("Metric: gdd=",gdd)
  print("Connection: GAMUDD=",GAMUDD)
  print("Riemann tensor RUDDD=",RUDDD)
  print("Ricci tensor RDD=",RDD)
  print("Ricci Curvature R=",R)
  f=quote(f)
  print("LaplF(f(X),X)=",LaplF(f(X),X))
  
  And smib gives:
  
  Sol Metric: gdd= ((exp(2*x),0,0),(0,exp(-2*x),0),(0,0,1))
  Connection: GAMUDD=(((1,0,0),(0,exp(-4*x),0),(0,0,0)),
  ((0,-1,0),(-1,0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0)))
  Riemann tensor RUDDD=
  ((((0,0,0),(0,0,0),(0,0,0)),
  ((0,-2*exp(-4*x),0),(2*exp(-4*x),0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0))),
  (((0,2,0),(-2,0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0))),
  (((0,0,0),(0,0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0)),
  ((0,0,0),(0,0,0),(0,0,0))))
  Ricci tensor RDD= ((-2,0,0),(0,-2*exp(-4*x),0),(0,0,0))
  Ricci Curvature R= -4*exp(-2*x)
  LaplF(f(X),X)= d(d(f((x,y,z)),z),z)+exp(-2*x)*d(d(f((x,y,z)),x),x)
  -2*exp(-2*x)*d(f((x,y,z)),x)+exp(2*x)*d(d(f((x,y,z)),y),y)
• ↓↓ [J]  [J] Karl Schwarzschild (1873-1916)Schwarzschild metric:
  X=(x1,x2,x3,x4) gdd=((-x1/(x1-a),0,0,0),(0,-x1^2,0,0),
  (0,0,-x1^2*sin(x2)^2,0),(0,0,0,-(x1-a)/x1))
  print("gdd=",gdd)
  riemann(gdd,X)
  print("GAMUDD[2,1,2]=",simplify(GAMUDD[2,1,2]))
  print("GAMUDD[3,1,3]=",simplify(GAMUDD[3,1,3]))
  print("RDD=",simplify(RDD))
  print("R=",simplify(R))
  
  smib gives:
  gdd= ((-x1/(-a+x1),0,0,0),(0,-x1^2,0,0),
  (0,0,-x1^2*sin(x2)^2,0),(0,0,0,-1+a/x1))
  GAMUDD[2,1,2]= 1/x1
  GAMUDD[3,1,3]= 1/x1
  RDD= 0
  R= 0
• The following function allows the testing of some properties of tensors and covariant derivative↓↓.

5.5 ↓Numerical analysis

5.5.1 Sample↓s applied to numerical analysis

5.5.2 Special functions

5.5.3 Numerical interpolation

5.5.4 Numerical integration

↓Integrals on compact

• ∫¹₀√(x)dx, ∫¹₀x^3 ⁄ 2dx
• ∫¹₀¹⁄_1 + xdx
• ∫¹₀¹⁄_1 + x⁴dx
• ∫¹₀¹⁄_1 + e^xdx
• ∫¹₀^x⁄_{− 1 + e^x}dx
• ∫¹₀²⁄_{2 + sin(10πx)}dx
• ∫¹₀¹⁄_{x^1⁄2 + x^1⁄3}dx.

↓↓Oscillatory integrals

• I₀(n) = ∫^2π₀xcos(x)sin(nx)dx
• I₁(n) = ∫^2π₀xcos(50x)sin(nx)dx
• I₂(n) = ∫^2π₀^xsin(nx)⁄_{√(1 − (^x⁄2π)²)}dx
• I₃(n) = ∫^2π₀ln(x)sin(nx)dx.
  

Integrals on semi-finite interval↓

• ∫^∞₀^e − x⁄_1 + x⁴dx
• ∫^∞₀e^− x2dx
• ∫^∞₀^e − x⁄_2x + 100dx
• ∫^∞₂¹⁄_xln(x)²dx
• ∫^∞₂¹⁄_xln(x)^3⁄2dx
• ∫^∞₂¹⁄_x^1.01dx
• ∫^∞₂^sin(x)⁄_xdx
• ∫^∞₂cos(π^x2⁄₂)dx
• ∫^∞₂e^− x2dx
• ∫^∞₂^{sin(x − 1)}⁄_{√(x(x − 2))}dx
• ∫^∞₂^x⁄_{e^x − 1}dx.
  

↓Oscillatory integrals on semi-finite interval

• I(w) = ∫^∞₀e^− xsin(wx)dx
• J(w) = ∫^∞₀sin(wx)^x⁄_1 + x²dx
• K(w) = ∫^∞₀sin(wx)^{sin(x⁄₂)2}⁄_xdx
• ∫^∞₀10e^{− xsin(16πx)}⁄_1 + x²dx.
  

↓↓Miscellanous integrals

• ∫¹₀e^{(x2(1 − x2))}dx
• I(t, n) = ∫^π₀^{cos(tsin(x) − nx)}⁄_πdx
  

↓Wavelets

5.5.5 ↓↓Discrete Fourier transform

• i^df(t)⁄_dt + f(t) = δ(t) (I)
• ^d2f(t)⁄_dt² − f(t) = δ(t) (II)

5.5.6 Complex analysis↓

5.5.7 ↓Ordinary differential equation

5.5.8 ↓Fractional calculus

• constant function f = 1, with q = ¹⁄₂:
  symbolic solution is equal to ¹⁄_√(πx), using this script:
  
  q=1/2
  x0=0
  N=20
  f=1
  gnuplot2D(sample(numfracder1(f,x,q,x0,a,N),a,1,10,50))
  quit()
  
  qgfe gives:
  
• constant function f = 1, with q = ^− 1⁄₂:
  symbolic solution is equal to 2√(^x⁄_π), using this script:
  
  q=-1/2
  x0=0
  N=20
  f=1
  gnuplot2D(sample(numfracder1(f,x,q,x0,a,N),a,1,10,50))
  quit()
  
  qgfe gives:
  
• function f = arctan(√(x)), with q = ¹⁄₂:
  symbolic solution is equal to ^{√(π⁄_1 + x)}⁄₂, using this script:
  
  q=1/2
  x0=0
  N=20
  f=arctan(sqrt(x))
  gnuplot2D(sample(numfracder1(f,x,q,x0,a,N),a,1,10,50))
  quit()
  
  qgfe gives:
  

• On constant function:
  
  f = 1, with q = ¹⁄₂: symbolic solution is equal to ¹⁄_√(πx), using this script:
  
  f2=1
  df12=1/sqrt(pi*x)
  df22=2*sqrt(x/pi)
  q=1/2
  f=f2
  df=df12
  A=sample(numfracder1(f,x,q,x0,a,N),a,1,10,50)
  B=sample(numfracder2(f,x,q,x0,a,N),a,1,10,50)
  DF=sample(df,x,1,10,50)
  print("Gap using 1st method: ",Snorm(Sdiff(A,DF)))
  print("Gap using 2nd method: ",Snorm(Sdiff(B,DF)))
  quit()
  
  smib gives:
  
  Gap using 1st method: (0.0396955,0.00525836,0.00180176)
  Gap using 2nd method: (0.0396955,0.00525836,0.00180176)
  
  constant function f = 1, with q = ^− 1⁄₂: symbolic solution is equal to 2√(^x⁄_π), using this script:
  
  f2=1
  df12=1/sqrt(pi*x)
  df22=2*sqrt(x/pi)
  q=-1/2
  f=f2
  df=df22
  A=sample(numfracder1(f,x,q,x0,a,N),a,1,10,50)
  B=sample(numfracder2(f,x,q,x0,a,N),a,1,10,50)
  DF=sample(df,x,1,10,50)
  print("Gap using 1st method: ",Snorm(Sdiff(A,DF)))
  print("Gap using 2nd method: ",Snorm(Sdiff(B,DF)))
  quit()
  
  And smib gives:
  
  Gap using 1st method: (0.117733,0.0159192,0.00428559)
  Gap using 2nd method: (0.117733,0.0159192,0.00428559)
  
  For the constant function, there is no gain: it is not surprising because f is constant, and we only modify integration scheme.
• On non constant function:
  
  function f = arctan(√(x)), with q = ¹⁄₂: symbolic solution is equal to ^{√(π⁄_1 + x)}⁄₂, using this script:
  
  q=1/2
  x0=0
  N=20
  f1=arctan(sqrt(x))
  df1=sqrt(pi/(1+x))/2
  df2=sqrt(pi)*(sqrt(1+x)-1)
  f=f1
  df=df1
  A=sample(numfracder1(f,x,q,x0,a,N),a,1,10,50)
  B=sample(numfracder2(f,x,q,x0,a,N),a,1,10,50)
  DF=sample(df,x,1,10,50)
  print("Gap using 1st method: ",Snorm(Sdiff(A,DF)))
  print("Gap using 2nd method: ",Snorm(Sdiff(B,DF)))
  quit()
  
  Then we have:
  
  Gap using 1st method: (0.0181275,0.00253013,0.000204211)
  Gap using 2nd method: (0.00925736,0.00129616,1.80853e-05)
  
  function f = arctan(√(x)), with q =  − ¹⁄₂: symbolic solution is equal to √(π)(√(1 + x) − 1), using this script:
  
  q=-1/2
  x0=0
  N=20
  f1=arctan(sqrt(x))
  df1=sqrt(pi/(1+x))/2
  df2=sqrt(pi)*(sqrt(1+x)-1)
  f=f1
  df=df2
  A=sample(numfracder1(f,x,q,x0,a,N),a,1,10,50)
  B=sample(numfracder2(f,x,q,x0,a,N),a,1,10,50)
  DF=sample(df,x,1,10,50)
  print("Gap using 1st method: ",Snorm(Sdiff(A,DF)))
  print("Gap using 2nd method: ",Snorm(Sdiff(B,DF)))
  quit()
  
  Then we have:
  
  Gap using 1st method: (0.116059,0.0157821,0.00387766)
  Gap using 2nd method: (0.0132159,0.00165729,0.000823485)
  
  function f = sin(√(x)), with q = ¹⁄₂: symbolic solution is equal to ^√(π)⁄₂J_o(x), in a similar way:
  
  Gap using 1st method: (7.20872,0.90126,0.454605)
  Gap using 2nd method: (7.21362,0.901883,0.454891)
  
  function f = sin(√(x)), with q =  − ¹⁄₂: symbolic solution is equal to √(πx)J₁(x):
  
  Gap using 1st method: (14.0716,1.53183,1.23937)
  Gap using 2nd method: (14.1149,1.54285,1.23535)

• f(x) = √(1 + x), q = ¹⁄₂, here derivative is: d^1⁄₂f(x) = ¹⁄_√(πx) + ^{arctan(√(x))}⁄_√(π)
  With the following script:
  
  print("sqrt(1+x), q=1/2")
  f=sample(sqrt(1+x),x,0,10,150)
  q=1/2
  qder(x)=num(Sfracd(f,q,x)-1/sqrt(pi*x)-arctan(sqrt(x))/sqrt(pi))
  print("Gap:",Snorm(sample(qder,x,1,10,150)))
  
  smib returns:
  
  sqrt(1+x), q=1/2
  Gap: (0.0612795,0.0034718,0.00357985)
• f(x) = J₀(√(x)), q =  − ¹⁄₂, here derivative is: d^{− 1⁄₂}f(x) = ^2sin(√(x))⁄_√(π)
  With the following script:
  
  print("besselj(sqrt(x),0), q=-1/2")
  f=sample(besselj(sqrt(x),0),x,0,10,150)
  q=-1/2
  qder(x)=num(Sfracd(f,q,x)-2*sin(sqrt(x))/sqrt(pi))
  print("Gap:",Snorm(sample(qder,x,1,10,150)))
  
  smib returns:
  
  besselj(sqrt(x),0), q=-1/2
  Gap: (0.0984872,0.00676304,0.00430091)
• f(x) = J₀(√(x)), q = ¹⁄₂, here derivative is: d^1⁄₂f(x) = ^cos(√(x))⁄_√(πx)
  With the following script:
  
  print("besselj(sqrt(x),0), q=1/2")
  f=sample(besselj(sqrt(x),0),x,0,10,150)
  q=1/2
  qder(x)=num(Sfracd(f,q,x)-cos(sqrt(x))/sqrt(pi*x))
  print("Gap:",Snorm(sample(qder,x,1,10,150)))
  
  smib returns:
  
  besselj(sqrt(x),0), q=1/2
  Gap: (0.0465338,0.00206661,0.00317324)
• For f(x) = 1, we want to verify d^1⁄₂(d^1⁄₂)f = df = 0:
  With the following script:
  
  N=170
  a=0
  b=50
  f1=1
  q=1/2
  print(" f=",f1,", q=",q,", a=",a,", b=",b,", N=",N)
  f=sample(f1,x,a,b,N)
  qder(x)=num(Sfracd(f,q,x))
  S1=sample(qder,x,a+1,b,N)
  print(" f=",f1,", q=1")
  qder2(x)=num(Sfracd(S1,q,x))
  S2=sample(qder2,x,a+2,b,N)
  Sth=sample(d(f1,x),x,a+2,b,N)
  print("Err=",Snorm(Sdiff(S2,Sth)))
  gnuplot2D(Sdiff(S2,Sth))
  
  smib gives:
  
  f= 1 , q= 1/2 , a= 0 , b= 50 , N= 170
  f= 1 , q=1
  Err= (0.299963,0.00844421,0.021328)
  
  Here we plot error:
  
• For f(x) = 1, we want to verify d^{− 1⁄₂}(d^1⁄₂)f = f:
  With the following script:
  
  N=170
  a=0
  b=100
  f1=1
  q=1/2
  f=sample(f1,x,a,b,N)
  qder(x)=num(Sfracd(f,q,x))
  S1=sample(qder,x,a+1,b,N)
  qder2(x)=num(Sfracd(S1,-q,x))
  S2=sample(qder2,x,a+2,b,N)
  gnuplot2D(S2)
  
  smib gives: (here result is not very good...)
  

• For a sample, fractional derivative computation could not start at beginning of sample, because there is not enough informations.
• Number of sampling points could not exceed 170, because Γ function does not accept a greater argument (another ↓weakness).

5.6 Probalility↓ & statistic↓

5.6.1 ↓Distributions

• ↓↓bernoullidist(n,p)=(1-p)*dirac(n)+p*dirac(n-1)
• ↓↓Betabinomialdist(x,n,a,b)=Beta(x+a,n-x+b)*binomial(n,x)/Beta(a,b)
• ↓↓binomialdist(n,p,N)=binomial(N,n)*p^n*(1-p)^(N-n)
• ↓↓geometricdist(n,p)=p*(1-p)^n
• ↓↓hypergeometricdist(x,n,m,N)=binomial(n,i)*binomial(m,N-i)/binomial(n+m,N)
• ↓↓logseriesdist(n,x)=-x^n/(n*log(1-x))
• ↓↓multinomialdist(N,P)=dim(N)!*product(ind,1,dim(N),P[ind]^N[ind])/product(ind,1,dim(N),N[ind]!)
• ↓↓negativebinomialdist(x,p,r)=binomial(x+r-1,r-1)*p^r*(1-p)^x
• ↓↓ [Q]  [Q] Siméon Denis Poisson (1781–1840)poissondist(n,nu)=nu^n*exp(-nu)/n
• ↓↓Betadist(x,a,b)=carac(x,0,1)*(1-x)^(b-1)*x^(a-1)/Beta(a,b)
• ↓↓Betaprimedist(x,a,b)=heaviside(x)*(1+x)^(-a-b)*x^(a-1)/Beta(a,b)
• ↓↓ [R]  [R] Félix Édouard Justin Émile Borel (1871-1956)borel [S]  [S] Tanner (????-????)tannerdist(x)=1-sum(ind,1,prec,ind^(ind-2)*(x*exp(-x))^ind/(x*ind!))
• ↓↓ [T]  [T] Satyendranath Bose (1894-1974)boseeinsteindist(x,m,s)=x^s/(exp(x-m)-1)
• ↓↓cauchydist(x,b,m)=b/(pi*((x-m)^2+b^2))
• ↓↓chidist(x,n)=heaviside(x)*2^(1-n/2)*x^(n-1)*exp(-x^2/2)/Gamma(n/2)
• ↓↓chisquareddist(x,r)=heaviside(x)*x^(r/2-1)*exp(-x/2)/(Gamma(r/2)*2^(r/2))
• ↓↓ [U]  [U] Agner Krarup Erlang (1878-1929)erlangdist(x,h,l)=heaviside(x)*l*(l*x)^(h-1)*exp(-l*x)/(h-1)!
• ↓↓exponentialdist(x,l)=heaviside(x)*l*exp(-l*x)
• ↓↓extremevaluedist(x,a,b)=exp((a-x)/b-exp((a-x)/b))/b
• ↓↓Fdist(x,m,n)=heaviside(x)*Gamma((m+n)/2)*(m*x/n)^(m/2-1)*(1+m*x/n)^(-(m+n)/2)/(Gamma(m/2)*Gamma(n/2))
• ↓↓ [V]  [V] Enrico Fermi (1901-1954)fermi [W]  [W] Paul Adrien Maurice Dirac (1902-1984)diracdist(x,m,s)=x^s/(exp(x-m)+1)
• ↓↓zdist(x,n1,n2)=heaviside(x)*2*n1^(n1/2)*n2^(n2/2)*exp(n1*x)/(Beta(n1/2,n2/2)*(n1*exp(2*x)+n2)^((n1+n2)/2))
• ↓↓Gammadist(x,h,l)=heaviside(x)*l*(l*x)^(h-1)*exp(-l*x)/(h-1)!
• ↓↓ [X]  [X] Robert Gibrat (1904-1980)gibratdist(x)=heaviside(x)*exp(-log(x)^2/2)/(x*sqrt(2*pi))
• ↓↓halfnormaldist(x,t)=heaviside(x)*2*t*exp(-(x*t)^2/pi)/pi
• ↓↓inversegaussiandist(x,l,m)=heaviside(x)*sqrt(l/(2*pi*x^3))*exp(-l*(x-m)^2/(2*x*m^2))
• ↓↓laplacedist(x,b,m)=exp(-abs(x-m)/b)/(2*b)
• ↓↓lognormaldist(x,m,s)=heaviside(x)*exp(-(log(x)-m)^2/(2*s^2))/(s*x*sqrt(2*pi))
• ↓↓logarithmicdist(x,a,b)=heaviside(x)*log(x)/(b*(log(b)-1)-a*(log(a)-1))
• ↓↓logisticdist(x,b,m)=exp(-(x-m)/b)/(b*(1+exp(-(x-m)/b))^2)
• ↓↓ [Y]  [Y] James Clerk Maxwell (1831-1879)maxwelldist(x,a)=sqrt(2/pi)*x^2*exp(-x^2/(2*a^2))/a^3
• ↓↓normaldist(x,m,s)=exp(-(x-m)^2/(2*s^2))/(s*sqrt(2*pi))
• ↓↓ [Z]  [Z] Vilfredo Federico Damaso Pareto (1848-1923)paretodist(x,a,b)=heaviside(x)*a*b^a/x^(a+1)
• ↓↓ [A]  [A] John William Strutt, 3rd Baron Rayleigh (1842-1919)rayleighdist(x,s)=heaviside(x)*x*exp(-x^2/(2*s^2))/s^2)
• ↓↓ [B]  [B] William Sealy Gosset (1876-1937)studenttdist(x,n)=Gamma((n+1)/2)*(1+x^2/n)^((n+1)/2)/(sqrt(pi*n)*Gamma(n/2))
• ↓↓studentzdist(x,n)=Gamma(n/2)*(1+x^2)^(-n/2)/(sqrt(pi)*Gamma((n+1)/2))
• ↓↓uniformdist(x,a,b)=carac(x,a,b)/(b-a)
• ↓↓ [C]  [C] Richard Edler von Mises (1883-1953)vonmisesdist(x,a,b)=exp(b*cos(x-a))/besseli(b,0)
• ↓↓ [D]  [D] Ernst Hjalmar Waloddi Weibull (1887-1979)weibulldist(x,a,b)=a*b^(-a)*x^(a-1)*exp(-(x/b)^a)

5.6.2 ↓↓Distribution function

5.6.3 ↓Moment, ↓variance

5.6.4 Skewness↓, kurtosis↓

5.6.5 ↓Linear least squares fitting

5.6.6 ↓Median & quantile↓

5.6.7 Some examples

• What is the probability that 3 people will be ill ?
• What is the probability that at least 2 people will be ill ?

• mean = ¹⁄_σ
• variance =¹⁄_σ²

5.6.8 Random number generation

5.6.9 Classical approach using brownian motion ↓

5.6.10 stochastic differential equation↓↓↓

• ↓Euler- [F]  [F] Gisiro Maruyama (1916-1986)Maruyama method↓: X_t≃X_r + f(r, X_r)(t − r) + g(r, X)(B_t − B_r)
  So in smib:

•  [G]  [G] Grigori N. Milstein (????-????)Milstein method↓↓: X_t = X_r + f(r, X_r)(t − r) + g(r, X_r)(B_t − B_r) + ∂_xg(r, X_r)g(r, X_r)((B_t − B_r)² − (t − r))/(2)
  So in smib:

•  [H]  [H] Leonard Salomon Ornstein (1880-1941)Ornstein- [I]  [I] George Eugene Uhlenbeck (1900-1988)Uhlenbeck process↓↓: dX_t =  − aX_tdt + σdB_t, X₀ = x₀

• ↓ [J]  [J] Fischer Sheffey Black (1938-1995)Black- [K]  [K] Myron Samuel Scholes (1941-????)Scholes- [L]  [L] Robert C. Merton (1944-????)Merton process↓: dX_t = μX_tdt + σX_tdB_t, X₀ = x₀

5.6.11 Stratonovich calculus & Ito calculus

5.6.12 Stochastic differential equation : generalized approach using inversion method

5.6.13 In higher dimension

• \strikeout off\uuline off\uwave off↓↓Duffing-Van der Pool oscillator

• ↓↓Bonhoeffer Van der Pool oscillator

5.6.14 Partial differential equation↓↓ simulation using stochastic differential equation

5.7 Quantum mechanic↓

5.7.1 Basic quantum mechanic

1. they are norm preserving: If Û|ψ⟩ = |φ⟩, then ⟨ψ|ψ⟩ = ⟨φ|φ⟩,
2. a result which follows directly from (1): If |ψ⟩ is normalized, then |φ⟩ = Û|ψ⟩ is also normalized. This is the crucial property we use in quantum mechanic we require the symmetry operators to map the states (normalized vectors) to other states and this condition implies that the symmetry operators are unitary (we also need the conditions of linearity and invertibility),
3. they are inner product preserving: If |φ⟩ = Û|ψ⟩ and |φ₁⟩ = Û|ψ₁⟩, then ⟨φ|φ₁⟩ = ⟨ψ|ψ₁⟩,
4. they map orthonormal basis into orthonormal basis: If the set {..., ψ_i, ...} is an orthonormal basis for the Hilbert space and |φ_i⟩ = Û|ψ_i⟩ for all i, then {..., φ_i, ...} is also an orthonormal basis.

• 1st measurement of observable  : state |a_j⟩ with probability |⟨a_j|ψ⟩|²
• immediate 2nd measurement : state |a_j⟩ with probability 1.

• ℏ = ^h⁄_2π
• scalar complex conjugate a : conj(a)
• operator or A = ⎛⎜⎝ a₁₁ a₁₂     a₂₁ a₂₂ ⎞⎟⎠ : A=((a11,a12),(a21,a22))
• vector ket |a⟩ : a = (a1, a2, ..., an )
• vector bra ⟨a|: conj(a)=(conj(a1), conj(a2),...,conj(an))
• inner product ⟨a|b⟩: dot(conj(a),b)
• magnitude of vector |a⟩ : abs(a)
• normalize vector |a⟩ : norm(a)=normalize(a)=a/float(sqrt(dot(conj(a),a)))
• outer product |a⟩⟨b| : dot(a,conj(b))
• tensor product |x⟩⊗|y⟩ :
  Tprod(x,y)=prog(j,k,n,z,do(
  n=0, z=zero(dim(x)*dim(y)),
  for(j,1,dim(x),(for(k,1,dim(y),do(n=n+1,z[n]=x[j]*y[k])))),
  z
  ))
• complex conjugate of operator Â: conj(A)
• transpose of matrix A : transpose(A)
• Hermitian conjugate (adjoint) of operator A (A^⋆ = A^T): hconj(A)=transpose(conj(x))
• identity operator Î : unit(n)
• inner product ⟨a|A|b⟩ : dot(conj(a),dot(A,b))
• probability amplitude : pr(ai,psi)=float(mag(rect(dot(hconj(ai),psi)))^2)

5.7.2 Some examples

• photon crossing a polarizer

• linearly polarized photon crossing a polarizer

5.7.3 Quantum mechanic & Hermite function

• ↓creation: a⁺(f)(x) = ^{xf(x) − df(x)⁄_dx}⁄_√(2),
• ↓annihilation: a(f)(x) = ^{xf(x) + df(x)⁄_dx}⁄_√(2),
• ↓position: X(f)(x) = ^{√(2)(a(f(x)) + a + (f, x))}⁄₂,
• ↓impulsion: P(f)(x) = ^{i√(2)( − a(f)(x) + a + (f)(x))}⁄₂,
• ↓quanta number: N(f)(x) = a⁺(a(f))(x),
• ↓hamiltonian: H(f)(x) = N(f)(x) + ^f(x)⁄₂,
• ↓commutator: [a₁, a₂](f, x) = (a₁○a₂ − a₂○a₁)(f)(x).
  

5.8 Miscellaneous

5.8.1 Some logic : propositional logic↓

• max(x, y) = ⎧⎨⎩ x  x > y        y  otherwise    in smib: max(x,y)=test(x>y,x,y)
• min(x, y) = ⎧⎨⎩ y  x > y        x  otherwise    in smib: min(x,y)=test(x>y,y,x)
• logical negation↓↓: ¬x = 1 − x  in smib: Not(x)=1-x
• logical disjunction↓↓: x∨y = max(x, y)  in smib: Or(x,y)=max(x,y)
• logical conjunction↓↓: x∧y = min(x, y)  in smib: And(x,y)=min(x,y)
• ↓↓logical implication: x ⇒ y = ¬x∨y  in smib: Implies(x,y)=Or(Not(x),y)
• logical equality↓↓: x ⇔ y = (x ⇒ y)∧(y ⇒ x)  in smib:
  Equals(x,y)=And(Or(x,Not(y)),Or(Not(x),y))
• logical NAND↓↓: x⊗y = ¬x∧y  in smib: Nand(x,y)=Not(And(x,y))
• logical exclusive disjunction↓↓↓: x⊕y = (x∧¬y)∨(y∧¬x)  in smib:
  Xor(x,y)=Or(And(x,Not(y)),And(Not(x),y))

5.8.2 60 seconds in a minute, why ?

5.8.3 ↓↓Weyl sum

5.8.4 Graphs↓

• order of graph↓ ↓ o: number of vertices,
• ↓↓size of graph s: number of edges,
• density of graph↓↓ d = ^S⁄_o²,
• ↓↓adjacency matrix A,
• ↓number of triangles  = ^Tr(A3)⁄₆
• ↓↓degree matrix D,
• ↓↓laplacian matrix L = D − A.

• On last example, we see that eigen function may give erratic result...
• programs are not very difficult cf. init.cpp.

5.8.5 Quaternion↓

1. Qsum(Qneg(H), H) = 0,
2. Qsum(Qneg(H), H) = 0,
3. Qprod(Qinv(H), H) = (1, 0, 0, 0),
4. Qprod(H, Qinv(H)) = (1, 0, 0, 0),
5. Qcomm(Qinv(H), H) = 0,
6. Qcomm(H, Qinv(H)) = 0,
7. Associativity↓ of the three operations.
   

5.8.6 On ↓nonassociativity and Jacobi identity↓

• in ℝ³ with flat metric :
  
  ℝ³ is an eucliean vector space, we can define on this space the vector cross product↓↓. Let U = ⎛⎜⎜⎜⎝ u_x   u_y   u_z ⎞⎟⎟⎟⎠ and V = ⎛⎜⎜⎜⎝ v_x   v_y   v_z ⎞⎟⎟⎟⎠, the cross product is defined by U × V = ⎛⎜⎜⎜⎝ u_yv_z − u_zv_y   u_zv_x − u_xv_z   u_xv_y − u_yv_x ⎞⎟⎟⎟⎠ (in smib we use Levi-Civita symbol↓↓ crossP(U,V)=contr(contr(outer(LeviCivita(dim(U)),outer(U,V)),3,5),2,3)), we can show that the cross product is bilinear (i.e. (aU₁ + bU₂) × V = aU₁ × V + bU₂ × V and U × (aV₁ + bV₂) = aU × V₁ + bU × V₂). We want to verify :
  • U × (V × W) − (U × V) × W ≠ 0 : non-associativity
  • (U × V) × W + (V × W) × U + (W × U) × V = 0 : Jacobi identity
  So in smib :
  
       U1=(ux1,uy1,uz1)
       U2=(ux2,uy2,uz2)
       U3=(ux3,uy3,uz3)
       test(crossP(crossP(U1,U2),U3)-crossP(U1,crossP(U2,U3))==0,
       print("Associativity : Ok"),
       print("Associativity : Ko"))
       test(crossP(crossP(U1,U2),U3)+crossP(crossP(U2,U3),U1)+crossP(crossP(U3,U1),U2)==0,
       print("Jacobi identity : Ok"),
       print("Jacobi identity : Ko"))
  
  And smib gives :
  
       Associativity : Ko
       Jacobi identity : Ok
• In ℝ⁴ with non-flat metric↓↓ :
  We consider the following symmetric metric↓↓ g = ⎛⎜⎜⎜⎜⎜⎝ f₁ 0 0 0         0 f₂ 0 0         0 0 f₃ 0         0 0  0 f₄ ⎞⎟⎟⎟⎟⎟⎠on , [u, v] = ∇_vu − ∇_uv is the commutator↓ which is also bilinear. We want to verify :
  • [U, [V, W]] − [[U, V], W] ≠ 0 : non-associativity
  • [[U, V], W] + [[V, W], U] + [[W, U], V] = 0 : Jacobi identity
  So in smib :
       X=(x0,x1,x2,x3)
       W=(w0(X),w1(X),w2(X),w3(X))
       V=(v0(X),v1(X),v2(X),v3(X))
       U=(u0(X),u1(X),u2(X),u3(X))
       symetricmetric= ((f1(X), 0, 0, 0),
                    (0, f2(X), 0, 0),
                    (0, 0, f3(X), 0),
                    (0, 0, 0, f4(X)))
       riemann(symetricmetric,X)
       test(Commutator(Commutator(U,V),W)-Commutator(U,Commutator(V,W))==0,
            print("Associativity : Ok"),
            print("Associativity : Ko"))
       test(Commutator(Commutator(U,V),W)+Commutator(Commutator(V,W),U)+
           Commutator(Commutator(W,U),V)==0,
           print("Jacobi identity : Ok"),
           print("Jacobi identity : Ko"))
  
  And smib gives :
  
       Associativity : Ko
       Jacobi identity : Ok

5.8.7 ↓↓Gould number

5.8.8 ↓π computation (ten lessons for cooking π)

1. ↓↓ [A]  [A] John Wallis (1616-1703)Wallis formula: π = 2∏^j = ∞_j = 1^(2j2)⁄_{(2j − 1)(2j + 1)}
2. From Wallis formula: π = ∑^j = ∞_j = 0^{2j + 1j!2}⁄_{(2j + 1)!}
3. Unknown man formula: π = 4(1 − ∑^j = ∞_j = 1²⁄_{16j² − 1})
4. ↓ [B]  [B] Wang (????-????)Wang formula↓: π = 4(183arctan(¹⁄₂₃₉) + 32arctan(¹⁄₁₀₂₃) − 68arctan(¹⁄₅₈₃₂) + 
   12arctan(¹⁄₁₁₀₄₄₃) − 12arctan(¹⁄_4841182) − 100arctan(¹⁄_6826318))
5. Using ζ ↓↓Riemann function: π = √(6∑^j = ∞_j = 1¹⁄_j²)
6. Using sequence of prime numbers↓↓: π = √(6∏_j ∈ P¹⁄_{1 − ¹⁄j²}), where P is the set of prime numbers.
7. ↓↓ [C]  [C] Srinivâsa Aiyangâr Râmânujan (1887-1920)Ramanujan formula: ¹⁄_π = ^2√(2)⁄₉₈₀₁∑^j = ∞_j = 1^{(4j)!(1103 + 26390j)}⁄_{(j!⁴396^4j)}
8. ↓ [D]  [D] David Volfovich Chudnovsky (1947-????) & Gregory Volfovich Chudnovsky (1952-????)Chudnovsky formula↓: ¹⁄_π = 12∑^j = ∞_j = 0( − 1)^j(6j)!^{13591409 + 545140134j}⁄_{((3j)!j!³640320^{(3(j + 1 ⁄ 2))})}
9. ↓↓ [E]  [E] David Harold Bailey (1948-????)Bailey– [F]  [F] Peter Benjamin Borwein (1953-????)Borwein– [G]  [G] Simon Plouffe (1956-????)Plouffe formula: π = ∑^j = ∞_j = 016^− j[⁴⁄_8j + 1 − ²⁄_8j + 4 − ¹⁄_8j + 5 − ¹⁄_8j + 6]
10. ↓↓ [H]  [H] Fabrice Bellard (1972-????)Bellard formula: π = ∑^j = ∞_j = 0^{( − 1)j}⁄_2^10j[^− 25⁄_4j + 1 − ¹⁄_4j + 3 + ²⁸⁄_10j + 1 − ²⁶⁄_10j + 3
     − ²²⁄_10j + 5 − ²²⁄_10j + 7 + ¹⁄_10j + 9]

5.8.9 Electromagnetic tensor↓↓

• grad(f) = ∇f = (∂_xf, ∂_yf, ∂_zf),
• div(V) = ∇.V = ∂_xV_x + ∂_yV_y + ∂_zV_z,
• curl(V) = ∇ × V = ⎛⎜⎜⎜⎝ ∂_zV_y − ∂_yV_z   ∂_zV_x − ∂_xV_z   ∂_xV_y − ∂_yV_x ⎞⎟⎟⎟⎠,
• laplacian(f) = △f = ∂²_xf + ∂²_yf + ∂²_zf.

• F^μν =  − F^νμ
• F_μνF^μν = 2(B² − E²)
• det(F_μν) = (B.E)²
• ¹⁄₂ϵ_αβγδF^αβF^γδ =  − 4(B.E)

• F_αβ, γ + F_βγ, α + F_γα, β = 0
• F^αβ_, β = J^α

6 Short bibliography

• “Handbook of mathematical functions” (Abramovitz & Stegun) is a must read and a fabulous toolbox.
• On number theory:
  • “1001 problèmes en théorie classique des nombres” (Koninck & Mercier),
  • “Thèmes d’arithmétique“ (Bordelles),
  • “Algebra” (Lang),
  • “A=B” (Marko Petkovsek, Herbert Wilf & Doron Zeilberger).
• On analysis:
  • “Generalized Functions: Theory and Applications” (Kanwal),
  • “Eléments De Distributions Et D’équations Aux Dérivées Partielles” & “Problèmes de distributions avec solutions détaillées” (Zuily),
  • “Fourier transform” (Sneddon),
  • “Eléments De La Théorie Des Fonctions Et De L’analyse Fonctionnelle” (Kolmogorov & Fomine),
  • “A Treatise On Differential Equations” (Forsyth),
  • “Intégration et analyse de Fourier. Probabilités et analyse gaussienne” (Malliavin & Airault),
  • “Intégration Et Probalités, Analyse De Fourier - Exercices Corrigés” (Letac),
  • “Analyse continue par ondelettes” (Torrésani & Meyer),
  • “Distribution theory and transform analysis” (Zemanian),
  • “Calcul différentiel complexe” (Leborgne),
  • “The Fractional Calculus” (Oldham & Spanier),
  • “Methods of Numerical Integration” (Rabinowitz & Davis),
  • “Symbolic integration I” (Bronstein),
  • “Calcul formel” (Davenport, Siret & Tournier).
• On vector and tensor calculus, differential geometry:
  • “Schaum’s Outline of Vector Analysis” (Spiegel),
  • “Schaum’s Outline of Tensor Calculus” (Kay),
  • “Vector analysis” (Brand),
  • “Calcul tensoriel en physique” (Hladik),
  • “Calcul différentiel et géométrie” (Leborgne),
  • “Géométrie différentielle: variétés, courbes et surfaces” (Berger & Gostiaux).
• On probability & statistic:
  • “Probabilites Et Statistique - Cours Et Problèmes” (Spiegel),
  • “Les Probabilités Et La Statistique De A à Z” (Dress),
  • “Introduction au calcul stochastique appliqué à la finance” (Lamberton & Lapeyre),
  • “Stochastic Simulation: Algorithms and Analysis” (Asmussen & Glynn).
  • “Méthodes de Monte-Carlo et processus stochastiques : du linéaire au non-linéaire” (Gobet)
  • “Introduction to stochastic analysis and Malliavin calculus” (Da Prato)
• Miscellanous:
  • “Atlas des Mathématiques” (Reinhardt & Soeder),
  • “Quantique” (Elbaz),
  • “Leçons sur la gravitation” (Feynman),
  • “Dictionnaire des mathématiques” (Bouvier, George & Le Lionnais).

7 Glossary

Nomenclature

Index

2D curves: ↑

2D plotting: ↑

3D curves: ↑

3D dynamic system: ↑

3D plotting: ↑

Airy functions: ↑

Archimedes constant: ↑

Bailey-Borwein-Plouffe formula: ↑

Bell number: ↑

Bellard formula: ↑

Bernoulli distribution: ↑

Bernoulli number: ↑, ↑

Bessel function of first kind: ↑

Bessel function of second kind: ↑

Bessel functions: ↑

Bessel modified function of first kind: ↑

Bessel modified function of second kind: ↑

Beta binomial distribution: ↑

Beta distribution: ↑

Beta function: ↑

Beta prime distribution: ↑

Bezout identity: ↑

Black-Scholes-Merton process: ↑, ↑

Bloch sphere: ↑

Bombieri conjecture: ↑

Bonhoeffer Van der Pool oscillator: ↑

Bonnet theorem: ↑

Borel-Tanner distribution: ↑

Bose-Einstein distribution: ↑

Box-Muller method: ↑

Box-Muller transform: ↑

Catalan number: ↑

Cauchy distribution: ↑

Christoffel symbol: ↑, ↑

Chudnovsky formula: ↑

Cox-Ingersoll-Ross process: ↑

De Morgan’s law: ↑

Dirichlet convolution product: ↑

Dirichlet product: ↑

Duffing-Van der Pool oscillator: ↑

Einstein tensor: ↑

Erlang distribution: ↑

Euler constant: ↑

Euler method: ↑

Euler numbers of first and second kind: ↑

Euler totient function: ↑

Euler-Maruyama method: ↑, ↑

F distribution: ↑

Fermat number: ↑

Fermi-Dirac distribution: ↑, ↑

Feynman-Kac formula: ↑

Fibonacci number: ↑, ↑

Fourier tranform of sample: ↑

Fourier transform: ↑

Fourier transform, continous: ↑

Fourier transform, discrete: ↑

Frobenius norm: ↑

Gamma distribution: ↑

Gamma function: ↑

Gauss equations: ↑

Gauss scheme: ↑, ↑, ↑, ↑, ↑, ↑

Gauss theorem: ↑

Gauss theory of surface: ↑

Gauss-Newton method: ↑

Gauss-Ostrogradski theorem: ↑

Genocchi number: ↑, ↑

Gibrat distribution: ↑

Gosper algorithm: ↑

Gould number: ↑

Green function: ↑

Green theorem: ↑

Grünwald definition: ↑

Grünwald definition, modified version of the: ↑

Guderman function: ↑

Hankel function of first kind: ↑

Hankel function of second kind: ↑

Hermite function: ↑

Hermite polynomial: ↑, ↑

Hermitian operators: ↑

Hilbert space: ↑

Ito’s lemma: ↑

Jacobi identity: ↑

Jacobi symbol: ↑, ↑

Lagrange interpolation problem: ↑

Laguerre polynomial: ↑, ↑

Laplace distribution: ↑

Laplace operator: ↑, ↑

Legendre polynomial: ↑, ↑

Legendre symbol: ↑

Levi-Civita symbol: ↑

Liouville function: ↑

Lorenz system: ↑, ↑

Lotka-Volterra equation: ↑

Mainardi-Codazzi equations: ↑

Maxwell distribution: ↑

Mersenne number: ↑, ↑

Mertens function: ↑, ↑

Milstein method: ↑, ↑

Möbius function: ↑

NAND: ↑

Newton basis polynomials: ↑

Newton method: ↑

Nil: ↑

Ornstein-Uhlenbeck process: ↑, ↑

Pareto distribution: ↑

Poisson distribution: ↑

Poisson law: ↑

Ramanujan formula: ↑

Rayleigh distribution: ↑

Redheffer matrix: ↑

Ricci curvature: ↑, ↑

Ricci scalar: ↑

Ricci tensor: ↑, ↑

Riemann function: ↑

Riemann method: ↑

Riemann scheme: ↑

Riemann tensor: ↑, ↑

Riemannian geometry: ↑

Risch algorithm: ↑

Runge-Kutta method: ↑

Rössler system: ↑, ↑

Schrodinger equation: ↑

Schwarzschild metric: ↑

Simpson method: ↑

Simpson numerical scheme: ↑

Simpson scheme: ↑, ↑, ↑, ↑, ↑

Sol: ↑

Solovay-Strassen test: ↑

Stirling numbers of first and second kind: ↑

Stokes theorem: ↑

Stratonovich calculus: ↑

Student-t distribution: ↑

Student-z distribution: ↑

Syracuse problem: ↑

Taylor partial sum: ↑

Tchebychev polynomial: ↑

Vasicek process: ↑

Wallis formula: ↑

Wang formula: ↑

Weibull distribution: ↑

Weingarten operator: ↑

Weyl sum: ↑

Wilson formula: ↑

Yun algorithm: ↑

Yun square free factorization: ↑

absolute value: ↑

absolute value of a sample: ↑

acceleration vector: ↑

adjacency matrix: ↑

adjoint: ↑

algorithm, Gosper: ↑

algorithm, Yun: ↑

algorithm, euclidean: ↑

algorithm, extended euclidean: ↑

annihilation: ↑

antider: ↑, ↑

antiderivative: ↑, ↑

antidifference: ↑

antilogy: ↑

area integral: ↑

argument: ↑

argument of a sample: ↑

arithmetic equations: ↑

arithmetic function: ↑

associativity: ↑

binomial coefficient: ↑

binomial distribution: ↑

binormal vector: ↑

boolean indicators: ↑

bra: ↑

brownian motion: ↑, ↑

central limit theorem: ↑

characteristic polynomial: ↑

chi distribution: ↑

chi squared distribution: ↑

chinese remainder theorem: ↑

circle: ↑

coefficient, connection: ↑

coefficient, correlation: ↑

coin-tossing: ↑

commutator: ↑, ↑, ↑

compilation: ↑

complex analysis: ↑

complex numbers: ↑

complex path integral: ↑, ↑

condition: ↑

conjugate: ↑

conjunction: ↑

connected components, number of: ↑

connection coefficient: ↑, ↑

constant, Archimedes: ↑

constant, Euler: ↑

continous Fourier transform: ↑

contraposition: ↑

convolution product: ↑, ↑

convolution product of two samples: ↑

coordinate, orthogonal: ↑

correlation coefficient: ↑

covariant derivative: ↑, ↑

covariant derivative of tensor: ↑

creation: ↑

curl: ↑

curl of a vector field: ↑

curvature: ↑, ↑

curvature, Ricci: ↑, ↑

curvature, gaussian: ↑

curvature, mean: ↑

curvature, principal: ↑

curvedness: ↑

curvilinear integral: ↑

cyclotomic polynomial: ↑

cylinder: ↑

decomposition of a rational function: ↑

decomposition of rational function: ↑

definite integral of a sample: ↑

degree matrix: ↑

degree of a polynomial: ↑

denominator: ↑

density of graph: ↑

derangement: ↑

derivation of non-integer order: ↑

derivative: ↑

derivative of a sample: ↑

derivative, covariant: ↑, ↑

derivative, directed covariant: ↑

determinant: ↑, ↑

deterministic test: ↑

difference between two samples: ↑

differential equation: ↑, ↑

differential equation, higher order ordinary: ↑

differential equation, stochastic: ↑

differential equations, system of: ↑

differential geometry: ↑

differential operator: ↑

differintegration: ↑

dilatation: ↑

dimension of a vector: ↑

directed covariant derivative: ↑

discrete Fourier transform: ↑

discrete moment: ↑

discriminant: ↑

disjunction: ↑

disjunction, exclusive: ↑

distribution: ↑

distribution Fermi-Dirac: ↑

distribution function: ↑

distribution, Bernoulli: ↑

distribution, Beta: ↑

distribution, Beta binomial: ↑

distribution, Beta prime: ↑

distribution, Borel-Tanner: ↑

distribution, Bose-Einstein: ↑

distribution, Cauchy: ↑

distribution, Erlang: ↑

distribution, F: ↑

distribution, Fermi-Dirac: ↑

distribution, Gamma: ↑

distribution, Gibrat: ↑

distribution, Laplace: ↑

distribution, Maxwell: ↑

distribution, Pareto: ↑

distribution, Poisson: ↑

distribution, Rayleigh: ↑

distribution, Student-t: ↑

distribution, Student-z: ↑

distribution, Weibull: ↑

distribution, binomial: ↑

distribution, chi: ↑

distribution, chi squared: ↑

distribution, exponential: ↑, ↑

distribution, extreme value: ↑

distribution, geometric: ↑

distribution, half normal: ↑

distribution, inverse gaussian: ↑

distribution, log series: ↑

distribution, log-normal: ↑

distribution, logarithmic: ↑

distribution, logistic: ↑

distribution, multinomial: ↑

distribution, negative binomial: ↑

distribution, normal: ↑, ↑

distribution, uniform: ↑, ↑

distribution, von Mises: ↑

distribution, z: ↑

distributivity: ↑

divergence: ↑, ↑

divergence of a vector field: ↑

divide differences: ↑

division, euclidean: ↑

divisor, great common: ↑

divisors, number of: ↑, ↑

divisors, product over: ↑

divisors, sum over: ↑

eigenstates: ↑

eigenvalue of symmetric matrix: ↑

eigenvalues: ↑

eigenvector of symmetric matrix: ↑

electromagnetic tensor: ↑

electron spin: ↑

equality: ↑, ↑

equation, Gauss: ↑

equation, Lotka-Volterra: ↑

equation, Mainardi-Codazzi: ↑

equation, Schrodinger: ↑

equation, arithmetic: ↑

equation, heat: ↑, ↑

equation, higher order ordinary differential: ↑

equation, ordinary differential: ↑, ↑

equation, partial differential: ↑

equation, stochastic differential: ↑

equations, system of differential: ↑

error functions: ↑

euclidean algorithm: ↑

euclidean division: ↑

euclidian division: ↑

even: ↑

exclusive disjunction: ↑

expansion of expression: ↑

exponential: ↑

exponential distribution: ↑, ↑

exponential function: ↑

extended euclidean algorithm: ↑

extented greater common divisor: ↑

extreme value distribution: ↑

factorial: ↑

factorization: ↑

factorization in prime factors: ↑

factorization, square-free: ↑

field of rational functions: ↑

first form: ↑

first kind, Bessel function of: ↑

first kind, Bessel modified function of: ↑

first kind, Euler numbers of: ↑

first kind, Hankel function of: ↑

first kind, Stirling numbers of: ↑

floating point conversion: ↑

form, first: ↑

form, second: ↑

formula, Bailey-Borwein-Plouffe: ↑

formula, Bellard: ↑

formula, Chudnovsky: ↑

formula, Feynman-Kac: ↑

formula, Ramanujan: ↑

formula, Wallis: ↑

formula, Wang: ↑

formula, Wilson: ↑

fractional calculus: ↑

fractional derivative of a sample: ↑

function: ↑

function, Airy: ↑

function, Bessel: ↑, ↑

function, Bessel modified: ↑

function, Beta: ↑

function, Euler totient: ↑

function, Gamma: ↑

function, Green: ↑

function, Guderman: ↑

function, Hankel: ↑

function, Hermite: ↑

function, Liouville: ↑

function, Mertens: ↑, ↑

function, Möbius: ↑

function, Riemann: ↑

function, arithmetic: ↑

function, distribution: ↑

function, error: ↑

function, exponential: ↑

function, greater integer: ↑

function, hyperbolic trigonometric: ↑, ↑

function, hypergeometric: ↑

function, inverse hyperbolic trigonometric: ↑, ↑

function, inverse trigonometric: ↑, ↑

function, logarithmic: ↑, ↑

function, rational: ↑

function, smaller integer: ↑

function, trigonometric: ↑, ↑

function, von Mangoldt: ↑

functions acting on integers: ↑

gaussian curvature: ↑

gcd: ↑

geometric distribution: ↑

gradient: ↑

gradient of a scalar field: ↑

graph: ↑

graph, density of: ↑

graph, order of: ↑

graph, size of: ↑

great common divisor: ↑

greater common divisor: ↑, ↑

greater integer function: ↑

half normal distribution: ↑

hamiltonian: ↑

harmonic mean of divisors: ↑

heat equation: ↑, ↑

helix: ↑

higher order ordinary differential equation: ↑

histogram: ↑, ↑

hyperbolic trigonometric function: ↑

hyperbolic trigonometric functions: ↑

hypergeometric distribution: ↑, ↑

hypergeometric function: ↑

identity, Bezout: ↑

imaginary part: ↑

imaginary part of a sample: ↑

implication: ↑

impulsion: ↑

increasing power division: ↑

index of a path: ↑

index, shape: ↑

inequality: ↑

infinity: ↑

inner product: ↑

integer: ↑

integer part: ↑

integer, kernel of: ↑

integral: ↑, ↑, ↑

integral by parts: ↑

integral of a sample: ↑

integral on compact: ↑

integral on semi-finite interval: ↑

integral, area: ↑

integral, complex path: ↑, ↑

integral, curvilinear: ↑

integral, miscellanous: ↑

integral, oscillatory: ↑

integral, vectorial: ↑

integration by part: ↑

interactive mode: ↑

interpolation: ↑

interpolation, polynomial: ↑

inverse: ↑

inverse gaussian distribution: ↑

inverse hyperbolic trigonometric function: ↑

inverse hyperbolic trigonometric functions: ↑

inverse trigonometric function: ↑

inverse trigonometric functions: ↑

inversion method: ↑

inversion problem: ↑

inversion, modular: ↑

isinteger: ↑

isprime: ↑

kernel of integer: ↑

kernel of the operator: ↑

ket: ↑

kurtosis: ↑, ↑, ↑, ↑

laplacian: ↑, ↑

laplacian matrix: ↑

laplacian of a scalar field: ↑

laplacian of a vector field: ↑

law of large numbers: ↑

least common multiple: ↑

limit: ↑

linear least squares fitting: ↑

log series distribution: ↑

log-normal distribution: ↑

logarithmic distribution: ↑

logarithmic function: ↑, ↑

logic conjonction: ↑

logic disjonction: ↑

logic negation: ↑

logical NAND: ↑

logical conjunction: ↑

logical disjunction: ↑

logical equality: ↑

logical exclusive disjunction: ↑

logical implication: ↑

logical negation: ↑

logical operator: ↑

logistic distribution: ↑

loop: ↑

lowering indices: ↑

matrices: ↑

matrix, Redheffer: ↑

matrix, adjacency: ↑

matrix, degree: ↑

matrix, laplacian: ↑

mean: ↑

mean curvature: ↑

median: ↑, ↑

median of a sample: ↑

median, quantile of a: ↑

method, Box-Muller: ↑

method, Euler: ↑

method, Euler-Maruyama: ↑, ↑

method, Gauss-Newton: ↑

method, Milstein: ↑, ↑

method, Newton: ↑

method, Riemann: ↑

method, Runge-Kutta: ↑

method, Simpson: ↑

metric: ↑, ↑

metric, Nil: ↑

metric, Schwarzschild: ↑

metric, Sol: ↑

metric, non-flat: ↑

metric, statical-spherical: ↑

metric, symmetric: ↑

miscellanous integrals: ↑

mode, interactive: ↑

mode, script: ↑

model, predator-prey: ↑

modified version of the Grünwald definition: ↑

modular inversion: ↑

modulation: ↑

modus barbara: ↑

modus ponens: ↑

modus tollens: ↑

moment: ↑

multinomial distribution: ↑

multiple, least common: ↑

negation: ↑

negative binomial distribution: ↑

nil: ↑, ↑

non-flat metric: ↑

non-integer order, derivation: ↑

non-linear least squares mean: ↑

nonassociativity: ↑

noncommutativity: ↑

norm of sample: ↑

normal distribution: ↑, ↑, ↑

normal vector: ↑, ↑, ↑

nth component: ↑

number of connected components: ↑

number of distinct prime factors: ↑

number of divisors: ↑, ↑

number of singularities: ↑

number of triangles: ↑

number, Bell: ↑

number, Bernoulli: ↑, ↑

number, Catalan: ↑

number, Euler: ↑

number, Fermat: ↑

number, Fibonacci: ↑, ↑

number, Genocchi: ↑, ↑

number, Gould: ↑

number, Mersenne: ↑, ↑

number, Stirling: ↑

numbers, prime: ↑

numerator: ↑

numerical analysis: ↑

numerical integration scheme: ↑

numerical scheme of integration: ↑

observable: ↑

odd: ↑

operator: ↑, ↑

operator acting on function: ↑

operator, Laplace: ↑

operator, Weingarten: ↑

operator, differential: ↑

operator, kernel of the: ↑

operator, logical: ↑

operator, pseudo-differential: ↑

operator, symbol of: ↑

order of graph: ↑

ordinary differential equation: ↑

ordinary differential equation solver: ↑

orthogonal coordinate: ↑

orthogonal polynomial: ↑, ↑

oscillator, Bonhoeffer Van der Pool: ↑

oscillator, Duffing-Van der Pool: ↑

oscillatory integral: ↑

oscillatory integral on semi-finite interval: ↑

outer product: ↑

partial differential equation: ↑

partial product: ↑

partial sum: ↑

partial sum, Taylor: ↑

path in the complex plane: ↑

path, complex integral: ↑

path, index of a: ↑

perfect number: ↑

permutation: ↑

photon: ↑

pi computation: ↑

planar curves: ↑

plotting: ↑

polar coordinates: ↑

polynomial: ↑, ↑

polynomial equation: ↑

polynomial interpolation: ↑

polynomial, Hermite: ↑, ↑

polynomial, Laguerre: ↑, ↑

polynomial, Legendre: ↑, ↑

polynomial, Tchebychev: ↑

polynomial, characteristic: ↑

polynomial, cyclotomic: ↑

polynomial, orthogonal: ↑, ↑

polynomial, roots of a: ↑

polynomials: ↑

position: ↑

position vector: ↑

predator-prey model: ↑

primality test: ↑

primality tests: ↑

prime counting: ↑

prime number theorem: ↑

prime numbers: ↑

primes, product over: ↑

primes, sum over: ↑

principal curvature: ↑

probabilistic test: ↑

probalility: ↑

process, Black-Scholes-Merton: ↑, ↑

process, Cox-Ingersoll-Ross: ↑

process, Ornstein-Uhlenbeck: ↑, ↑

process, Vasicek: ↑

product of a scalar and a sample: ↑

product of matrix by vector: ↑

product of two samples: ↑

product over divisors: ↑, ↑

product over prime divisors: ↑

product over primes: ↑

product, Dirichlet: ↑

product, Dirichlet convolution: ↑

product, convolution: ↑

product, partial: ↑

product, vector cross: ↑

program: ↑

programming language: ↑

projection: ↑

projector: ↑

proposition: ↑

propositional logic: ↑

pseudo-aleatory rational: ↑

pseudo-differential operator: ↑

quadratic variation: ↑

quanta number: ↑

quantile: ↑, ↑, ↑, ↑

quantile of a sample: ↑

quantum mechanic: ↑

quaternion: ↑

quaternion commutator: ↑

quaternion conjugation: ↑

quaternion inverse: ↑

quaternion negation: ↑

quaternion norm: ↑

quaternion product: ↑

quaternion sum: ↑

raising indices: ↑

random number: ↑

rational function: ↑, ↑

rational function, decomposition of: ↑

rational function, decomposition of a: ↑

rationnal numbers: ↑

real numbers: ↑

real part: ↑

real part of a sample: ↑

recursive loop: ↑

reduction to same denominator: ↑

resultant: ↑

roots of a polynomial: ↑

sample: ↑, ↑, ↑

sample filtering: ↑

sample truncation: ↑

sample, Fourier tranform of: ↑

sample, absolute value of a: ↑

sample, argument of a: ↑

sample, definite integral of a: ↑

sample, derivative of a: ↑

sample, fractional derivative of a: ↑

sample, imaginary part of a: ↑

sample, integral of a: ↑

sample, norm of: ↑

sample, product of a scalar and a: ↑

sample, quantile of a: ↑

sample, real part of a: ↑

samples, difference between two: ↑

samples, product of two: ↑

samples, sum of two: ↑

scalar field, gradient of a: ↑

scalar field, laplacian of a: ↑

scheme, Gauss: ↑, ↑, ↑

scheme, Simpson: ↑

script mode: ↑

second form: ↑

second kind, Bessel function of: ↑

second kind, Bessel modified function of: ↑

second kind, Euler numbers of: ↑

second kind, Hankel function of: ↑

second kind, Stirling numbers of: ↑

shape index: ↑

simplification of expression: ↑

singularities, number of: ↑

size of graph: ↑

skewness: ↑, ↑, ↑, ↑

smaller integer function: ↑

speed vector: ↑

sphere: ↑

spherical coordinates: ↑

square free factorization: ↑

square-free factorization: ↑

statical-spherical metric: ↑

statistic: ↑

stochastic differential equation: ↑

stochastic vector: ↑

sum: ↑

sum of divisors at order r: ↑

sum of two samples: ↑

sum over divisors: ↑, ↑

sum over primes: ↑

sum, Weyl: ↑

sum, partial: ↑

sum, partial Taylor: ↑

support: ↑

symbol of operator: ↑

symbol, Christoffel: ↑, ↑

symbol, Jacobi: ↑, ↑

symbol, Legendre: ↑

symbol, Levi-Civita: ↑

symmetric metric: ↑

system of differential equations: ↑

system, 3D dynamic: ↑

system, Lorenz: ↑

system, Rössler: ↑

tangent vector: ↑, ↑

tautology: ↑

temporary program: ↑, ↑

tensor: ↑

tensor product: ↑

tensor, Einstein: ↑

tensor, Ricci: ↑, ↑

tensor, Riemann: ↑, ↑

tensor, covariant derivative of: ↑

tensor, electromagnetic: ↑

theorem, Bonnet: ↑

theorem, Gauss: ↑

theorem, Gauss-Ostrogradski: ↑

theorem, Green: ↑

theorem, Stokes: ↑

theorem, central limit: ↑

theorem, chinese remainder: ↑

torsion: ↑

torus: ↑

total number of prime factors: ↑

trace: ↑

transform, Box-Muller: ↑

translation: ↑

transposition: ↑

trigonometric function: ↑, ↑

trigonometric functions: ↑

truth table: ↑

unidimensional integral calculus: ↑

uniform distribution: ↑, ↑, ↑

unity exponential: ↑

variance: ↑, ↑

vector cross product: ↑

vector field, curl of a: ↑

vector field, divergence of a: ↑

vector field, laplacian of a: ↑

vector, acceleration: ↑

vector, binormal: ↑

vector, normal: ↑, ↑, ↑

vector, position: ↑

vector, speed: ↑

vector, tangent: ↑, ↑

vectorial calculus: ↑

vectorial integral: ↑

vectors: ↑

von Mangoldt function: ↑

von Mises distribution: ↑

wavelet: ↑, ↑

weakness: ↑, ↑, ↑, ↑, ↑

wronskian: ↑

z distribution: ↑

---