Appendix F

Units, Constants, and Notations

This appendix simply lists various fundamental units, physical constants, and notations used throughout the text, for easy reference.

Table F.1 shows the names, abbreviations, and values of the standard order-of-magnitude prefixes for units of measurement.

Table F.2 shows the various base units of measurement referred to in this thesis. Any of these of course may also appear together with any of the prefixes listed in table F.1.

Table F.3 shows the fundamental physical constants we use. The Planck length L_{P} is listed in the units table (table F.2), but it may also be considered to be a fundamental physical constant. It is sometimes hypothesized to be some sort of minimum length scale for physics.

Table F.4 gives our preferred, more mnemonic notation for comparing the asymptotic order of growth of functions.

Name Sym Val Name Sym Val
deka- D 10^{1}  deci- d 10^{−1}
hecto- h 10^{2} centi- c 10^{−2}
kilo- k 10^{3} milli- m 10^{−3}
mega- M 10^{6} micro- μ 10^{−6}
giga- G 10^{9} nano- n 10^{−9}
tera- T 10^{12} pico- p 10^{−12}
peta- P 10^{15} femto- f 10^{−15}
exa- E 10^{18} atto- a 10^{−18}

Table F.1: Unit magnitude prefixes.

Symbol Name  Measures  Some equivalences
Å  Ångstrom  length  10^{−10} m
B  byte  information  8 b
b  bit  information, entropy (ln 2) nat
C  Coulomb  electric charge
eV  electron Volt energy  1.60217733×10^{−19} J
g  gram  mass
Hz  Hertz  frequency  1/s
J  Joule  energy  1 N m
K  Kelvin  temperature  ∼ 1.38×10^{−23} J/nat
kg  kilogram  mass  1000 g
m  meter  length
N  Newton  force  1 kg m/s^{2}
nat nat  entropy  k_{B}, 1 b/ln 2
L_{P}  Planck length length  \sqrt{Gℏ/c^{3}} ≈ 1.616×10^{−35} m
s  second  time
V  Volt  electric potential  1 J/C

Table F.2: Some basic units of measurement used in this document.

Symbol Meaning  Some equivalences Approximate value
c  Speed of light  299792458 m/s
ϵ_{0}  Permittivity of  1/μ_{0}c^{2}  8.85418782×10^{−12} F/m

free space

h  Planck’s constant  6.6260755×10^{−34} J · s
ℏ  Reduced Planck’s  h/2π  1.05457267×10^{−34} J · s

constant

G Gravitational 6.67259×10^{−11} N m^{2}/kg

constant

k_{B}  Boltzmann’s constant  1 nat  1.3806513×10^{−23} J/K
σ_{SB} Stefan-Boltzmann  π^{2}k_{B}^{4}/60c^{2}ℏ^{3}  5.6704×10^{−8} J/s-K^{4}-m^{2}

constant

q_{e} Electron charge 1.60217733×10^{−19} C

magnitude

Table F.3: Fundamental physical constants used in this document.

Cryptic Our own
standard mnemonic Mathematical definition;
notation notation English explanation

f = Θ(g) f ∼ g ∃c_{1}, c_{2}, n_{0} > 0: ∀n ≥ n_{0} : 0 ≤ c_{1}g(n) ≤ f(n) ≤ c_{2}g(n); f is asymptotically proportional to g

f = O(g) f ≾ g ∃c, n_{0} > 0: ∀n ≥ n_{0} : 0 ≤ f(n) ≤ cg(n); f is asymptotically no more than proportional to g

f = Ω(g) f ≿ g ∃c, n_{0} > 0: ∀n ≥ n_{0} : 0 ≤ cg(n) ≤ f(n); f is asymptotically no less than proportional to g

f = o(g) f ≺ g ∀c > 0: ∃n_{0} > 0: ∀n ≥ n_{0} : 0 ≤ f(n) < cg(n); f is asymptotically strictly less than proportional to g

f = ω(g) f ≻ g ∀c > 0: ∃n_{0} > 0: ∀n ≥ n_{0} : 0 ≤ cg(n) < f(n); f is asymptotically strictly more than proportional to g

Table F.4: Asymptotic order-of-growth notation. In addition to reviewing the standard notation, we introduce a simplified notation that will be useful in some contexts.

390 APPENDIX F. UNITS, CONSTANTS, AND NOTATIONS