# (p.231) List of Symbols

# (p.231) List of Symbols

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) intervalwith

*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 Ωσ(£) σ-algebra generated by collection

*Ε*of subsets of Ω23 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*.*(p.232) ΑΔΒ*symmetric difference of*A*and*Β; ΑΔΒ = (A\B)*U*(B\A)*λ Lebesgue measure on (R, Έ) or parts thereofλα6 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 measurablefunction / : 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)(p.233) (/"(xo)) orbit of xo (under map /); (/»(*<,)) = (/(zo), /o/(*o), · · ·)

0

*ο*.

real, imaginary part of

conjugate, absolute value (modulus) of

arg *ζ* argument of

arg

and

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