N, No, Ζ set of positive integers, nonnegative integers, integers
Q,R+,R, C set of rational, positive real, real, complex numbers
[a, b) (half-open) interval
with a < b.
D\,D-2, etc. first, second, etc. significant decimal digit
mt h significant digit base 6, b > 2 log χ logarithm base 10 of
logarithm base b of
In χ natural logarithm (base e) of
Δ deviation (in percent) from first-digit law:
Δ = 100 ·
Φ A cardinality (number of elements) of finite set A.
U(a, b) random variable uniformly distributed on (a, b) with a < b.
S (decimal) significand function
[x\ largest integer not larger than
(x) fractional part of
A σ-algebra, on some non-empty set Ω
σ(£) σ-algebra generated by collection Ε of subsets of Ω
23 Borel σ-algebra on R or parts thereof
σ(/) σ-algebra generated by function / : Ω —> R § significand σ-algebra; § = R+ Π σ (S).
(Ω, Α, Ρ) probability space
Ac complement of A in set
A\B set of elements in A but not in B; A\B = An Bc.
λα6 normalized Lebesgue measure (uniform distribution) on ([α,5),Έ[α,5))
δω Dirac probability measure concentrated at
p(C) (natural) density ofCcN
1A indicator function of set A.
(Fn) sequence of Fibonacci numbers; (Fn) = (1,1, 2, 3, 5,…)
(pn) sequence of prime numbers; (pn) = (2, 3, 5, 7,11,…)
i.i.d. independent, identically distributed
X, Y,… (real-valued) random variables Ω —> R
K[X] expected (or mean) value of random variable X.
varX variance of random variable X with E[|X|] < +∞: varX = E[(X–EX)2 ]
Ρ probability measure on (R, IB), possibly random
Ρχ distribution of random variable X.
Fp, Fx distribution function of Ρ, Χ.
kth Fourier coefficient of probability Ρ on S[0,1):
Β Benford distribution on (R+, §)
Δοο deviation (100 χ sup-norm) from Benford's law:
Φ standard normal distribution function:
Vf, P o / " 1 probability measure on R induced by Ρ and measurable
function / : Ω –• R, via P; ( · ) = F(7^1 (·))
u.d. mod 1 uniformly distributed modulo one
a.a. (Lebesgue) almost all
a.s. almost surely, i.e., with probability one
y, \ increasing, decreasing (not necessarily strictly)
real, imaginary part of
conjugate, absolute value (modulus) of
arg ζ argument of
S unit circle in
spanQ^ rational span of Ζ C C;
Ck set of all k times continuously differentiable functions
C°° set of all smooth (i.e., infinitely differentiable) functions
Ng Newton map associated with differentiable function g.
~Rdxd set (linear space) of all real d χ d-matrices
Id d χ d-identity matrix
La set (linear space) of all linear observables on Rd x d
[A]jk entry in j t h row, kth column of
σ (A) spectrum (set of all eigenvalues) of
p(A) spectral radius of
Euclidean norm of
first, second, kth derivative of χ = x(t) with respect to t.
(Xn) converges in distribution to X.
(Xn) converges to X almost surel.
r.p.m. random probability measure
EP expectation of r.p.m. Ρ.
T + set of all positive terminating decimals:
P* finitely additive probability measure on Ω C M
end of PROOF
end of EXAMPLE