http://www.alumni.caltech.edu/~croft/notes.txt
David Wallace Croft
Compilation copyright released to the public domain by the author.
19980118

Miscellaneous

The SCA also attempts to create an atmosphere embodying those lost ideals
that are found in medieval romances: chivalry, courtesy, and honor. We
recreate the Middle Ages as they ought to have been: doing away with the
strife and pestilence and emulating the beauty, grace, chivalry and
brotherhood.
 The Society for Creative Anachronism (\SCA\PAMPHLET.SCA rev. 6/94)
Reminder of important documents to have ready in a secure place
(01) A current will
(02) Permanent file of military/business documents
(03) Birth certificates  yours, your spouse's, and dependents'
(04) Marriage certificates
(05) Health records  yours, your spouse's, and dependents'
(06) Income Tax returns  last four years
(07) Social Security Numbers (SSN)  yours, your spouses, and dependents'
(08) Real estate deeds, mortgages, and insurance
(09) Insurance papers  life and health
(10) Bank Accounts  type numbers and locations
(11) Stocks and bonds

American Heritage Dictionary, 2nd College Edition, 1982

elitism: n. 1.a. Belief in rule by an elite.
paroxysm: n. 1. A sudden outburst of emotion or action: a paroxysm
of laughter. 2. Pathol. a. A crisis in or recurrent
intensification of a disease. b. A spasm or fit; convulsion.
paternalism: n. A policy or practice of treating or governing
people in a fatherly manner, esp. by providing for their needs
without giving them responsibility.
pithy: adj. 2. Precisely meaningful; cogent and terse.
surmise: v. tr. To infer (something) without sufficiently
conclusive evidence; guess. intr. To make a guess or conjecture.

Algebra
Serway, Raymond A. Physics for Scientists & Engineers, 3/e, Volume 2,
Updated Printing. 1992.

The general form of a quadratic equation is
a * ( x ** 2 ) + b * x + c = 0
where x is the unknown quantity and a, b, and c are numerical
factors referred to as coefficients of the equation.
The quadratic equation has two roots, given by
___________________________
b ñ û ( b ** 2 )  ( 4 * a * c )
x = .
2 * a
The roots will be real in the quadratic equation, if
( b ** 2 ) >= ( 4 * a * c ).

Linear Algebra
(from the class notes of Caltech's CNS 185 "Collective Computation")

Multiplying m x n matrix A by n x p matrix B produces an m x p
matrix C = A * B whose elements are defined to be
n
C_ik = ä A_ij * B_jk.
j = 1
Matrix multiplication can only be performed between two matrices
A and B if (columns of A) = (rows of B).
Like ordinary multiplication, matrix multiplication is associative
and distributive, but unlike ordinary multiplication, it is not
commutative. A * B /= B * A
Transpose ( A * B ) = Transpose ( B ) * Transpose ( A )
The inner product (also known as the dot product) of ndimensional
vectors x and y is defined as Transpose ( x ) * y which is scalar.
The inner product is the sum of the products of corresponding
elements from the two vectors:
n
Transpose ( x ) * y = ä x_i * y_i.
i = 1
If the inner product of two vectors is zero, they are said to be
orthogonal, which has the usual geometric connotation of
perpendicularity.
The diagonal of an n x n square matrix A are the elements A_ii
running diagonally from the top left corner to the bottom right.
A diagonal matrix is a matrix which has zeroes everywhere off the
diagonal.
The symbol I is reserved for a particular diagonal matrix known as
the identity matrix, which has ones along its diagonal and zeroes
elsewhere. It is the multiplicative identity for matrix
multiplication of square matrices. A * I = I * A = A.
The n x n square matrix A is called invertible if there exists a
matrix denoted A ** 1 which satisfies
A * ( A ** 1 ) = ( A ** 1 ) * A = I.
If A ** 1 exits, it is called the inverse matrix.
If A ** 1 does not exist, A is called a singular matrix.
( Transpose ( A ) ) ** 1 = Transpose ( A ** 1 )
( A * B ) ** 1 = ( B ** 1 ) * ( A ** 1 )
All square matrices have a particular scalar value associated with
them, known as the determinant, which is written as: det A =  A .
For two dimensions, the formula for calculating the determinant is
 a_11 a_12 
det A =  A  =   = a_11 * a_12  a_12 * a_21.
 a_21 a_22 
If the determinant of a matrix is zero, the matrix is singular.
iff == "if and only if"
A simple way to find if a matrix A is invertible or not is to find
its determinant, since ( det A = 0 ) iff ( A is not invertible ).
If A is invertible, the only vector x satisfying A * x = 0 is x = 0.
If A is not invertible, there can be interesting nonzero solutions
for x.
If vector x satisfies A * x = lambda * x, so does à * x. So you won't
be able to solve for a unique x, just for the direction that x should
lie in. That direction is the eigenvector direction, and all vectors
parallel to it are eigenvectors with the same eigenvalue.

Feller, William. An Introduction to Probability Theory and Its
Applications. Vol I, 3rd Ed. 1960. Chapter 1.

A compound (or decomposable) event is an aggregate of certain simple
events. Example: the compound event of "sum of six" in rolling
2d6 (two 6sided dice) is composed of the 5 simple events
"(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)".
In set theory, "A * B = 0" means that A and B are mutually exclusive.
U == or, union, logical sum, +
ï == and, intersection, simultaneous realization, *
_
_ (sideways U) == "is contained in"
_
_ (sideways U) == "contains"
î == "is a member of"
_
A implies B == A _ B
_
B is implied by A == B _ A
Truth Table for "Implies": Z = A Imp B = A' + B
A B Z

0 0 1
0 1 1
1 0 0
1 1 1
A  A * B = A * B' = A occurs but not B
A sample space is called discrete if it contains only finitely many
points or infinitely many points which can be arranged into a simple
sequence E1, E2, ....
The probability of any event is between 0 and 1 inclusive:
0 <= P { A } <= 1
P { A1 U A2 } <= P { A1 } + P { A2 }
Boole's Inequality
P { A1 U A2 U A3 U ... } <= P { A1 } + P { A2 } + P { A3 } + ...
P { A1 U A2 } = P { A1 } + P { A2 }  P { A1 ï A2 }

Feller, William. An Introduction to Probability Theory and Its
Applications. Vol I, 3rd Ed. 1960. Chapter 2.

Pairs. With m elements a_1, ..., a_m and n elements b_1, ..., b_n,
it is possible to form m * n pairs ( a_j, b_k ) containing one element
from each group.
Multiplets. Given n_1 elements a_1, ..., a_n_1 and n_2 elements
b_1, ..., b_n_2, etc., up to n_r elements x_1, ..., x_n_r;
it is possible to from n_1 * n_2 * ... * n_r ordered rtuplets
( a_j1, b_j2, ..., x_jr ) containing one element of each kind.
Sampling without replacement notation
(n)_r = n * ( n  1 ) * ( n  2 ) * ... * ( n  r + 1 )
(n)_r = 0 for integers r, n such that r > n
For a population of n elements and a prescribed sample size r, there
exist n ** r different samples with replacement and (n)_r samples
without replacement.
In sampling without replacement a sample of size n includes the
whole population and represents a reordering (or permutation) of
its elements. The number of different orderings of n elements is
(n)_n = n * ( n  1 ) * ... * 2 * 1 = n!
Whenever we speak of random samples of fixed size r, the adjective
random is to imply that all possible samples have the same
probability, namely, n ** r in sampling with replacement and
1 / (n)_r in sampling without replacement, n denoting the size of
the population from which the sample is drawn.
If n is large and r relatively small, the ratio (n)_r / ( n ** r )
is near unity. This leads us to expect that, for large populations
and relatively small samples, the two ways of sampling are
practically equivalent.
In sampling without replacement the probability for any fixed element
of the population to be included in a random sample of size r is
1  ( n  1 )_r / ( n )_r = 1  ( n  r ) / n = r / n.
In sampling with replacement the probability that an element be
included at least once is 1  ( 1  ( 1 / n ) ) ** r.
A random sample of size r with replacement is taken from a
population of n elements. Assuming that all arrangements have equal
probability, we conclude that the probability of no repetition in
our sample is
p = (n)_r / ( n ** r ) = n * ( n  1 ) * ... * ( n  r + 1 ) / n ** r.
If n balls are randomly placed into n cells, the probability that
each cell will be occupied equals n! / n ** n. For n = 7, is is only
0.00612.
The probability that in a group of r people no one has the same
birthday as another is ( 365 )_r / 365 ** r. For 23 people the
probability that at least 2 people have a common birthday exceeds 0.5.
log ( 1  x ) ~= x
1 + 2 + 3 + ... + ( r  1 ) = r * ( r  1 ) / 2
We use the term "population of size n" to denote an aggregate of n
elements without regard to their order. Two populations are
considered different only if one contains an element not contained
in the other.
The standard notation for the binomial coefficients is
/ n \ n * ( n  1 ) * ... * ( n  r + 1 )
  = ( n )_r / r! = 
\ r / 1 * 2 * ... ( r  1 ) * r
("n choose r"), the number of subpopulations of size r in a
population of size n.
Theorem 1. A population of n elements possesses "n choose r"
different subpopulations of size r <= n. In other words,
a subset of r elements can be chosen in "n choose r" different ways.
"n choose r" = "n choose n  r" since there are as many subpopulations
of size r as there are subpopulations with ( n  r ) elements not
belonging to them.
/ n \ n! ( n )_r ( n  r )!
  =  =  * 
\ r / r! ( n  r )! r! ( n  r )!
n * ( n  1 ) * ... * ( n  r + 1 ) * ( n  r )!
= 
r! * ( n  r )!
"n choose 0" = 1
0! = 1
( n )_0 = 1
a "choose" b = 0 whenever b > a
x "choose" r = 0 whenever ( r < 0 ) or ( r > n )
Binomial Distribution: the probability that a specified cell contains
exactly k balls (k = 0, 1, 2, ..., r ) is
/ r \ 1
p_k =   *  * ( n  1 ) ** ( r  k )
\ k / ( n ** r )
/ r \ 1
=   *  * ( 1  1 / n ) ** ( r  k ).
\ k / ( n ** k )
Theorem 2. Let r_1, r_2, ..., r_k be integers such that
r_1 + r_2 + ... + r_k = n, r_i >= 0. The number of ways in which a
population of n elements can be divided into k ordered parts
(partitioned into k subpopulations) of which the first contains
r_1 elements, the seconds r_2 elements, etc., is
n! / ( r_1! * r_2! * ... * r_k! ). (multinomial coefficients)
The number of distinguishable distributions, that is,
the number of different solutions of r_1 + r_2 + ... + r_n = r, is
/ n + r  1 \ / n + r  1 \
A_r,n =   =  .
\ r / \ n  1 /
The number of distinguishable distributions in which no cell remains
empty is ( r  1 ) "choose" ( n  1 ). Example: there are
( r + 5 ) "choose" 5 distinguishable results of a throw with r
indistinguishable dice.
2d6 ==> 7 choose 5 = 7! / ( 5! * ( 7  5 )! ) = 7 * 6 / 2 = 21.
11
12 22
13 23 33
14 24 34 44
15 25 35 45 55
16 26 36 46 56 66
MaxwellBoltzmann statistics: ?
BoseEinstein statistics: 1 / A_r,n ?
FermiDirac statistics are based on these hypotheses: (1) it is
impossible for two or more particles to be in the same cell, and
(2) all distinguishable arrangements satisfying the first condition
have equal probabilities.
Misprints. A book contains n symbols (letters), of which r are
misprinted. The distribution of misprints corresponds to a
distribution of r balls in n cells with no cell containing more than
one ball. It is therefore reasonable to suppose that, approximately,
the misprints obey the FermiDirac statistics.
In any ordered sequence of elements of two kinds, each maximal
subsequence of elements of like kind is called a run. For example,
the sequence aaabaabbba opens with an "a" run of 3; it is followed by
runs of length 1, 2, 3, 1, respectively. The "a" and "b" runs
alternate so that the total number of runs is always one plus
the number of conjunctions of unlike neighbors in the given sequence.
Hypergeometric Distribution
In a population of n elements n_1 are red and n_2 = n  n_1 are black.
A group of r elements is chosen at random. The probability q_k that
the group so chosen will contain exactly k red elements is q_k =
= ( n_1 choose k ) * ( ( n  n_1 ) choose ( r  k ) ) / ( n choose r )
= ( r choose k ) * ( ( n  r ) choose ( n_1  k ) ) / ( n choose r )
For any particular set of observations n_1, r, k, the value of n for
which q_k ( n ) is largest is denoted by n^ and is called the
maximum likelihood estimate of n. This notion was introduced by
R. A. Fisher.
q_0 + q_1 + q_2 + ... = 1.
/ n \ / r \ / n  r \ / r \ / n  r \
  =   *   +   *   + ...
\ n_1 / \ 0 / \ n_1 / \ 1 / \ n_1  1 /
/ r \ / n  r \
+   *   for any positive integers n, n_1, and r.
\ n_1 / \ 0 /
Bridge. The population of 52 cards consists of four classes (suits),
each of 13 elements. The probability that a hand of 13 cards consists
of 5 spades, 4 hearts, 3 diamonds, and 1 club is
( 13 choose 5 ) * ( 13 choose 4 ) * ( 13 choose 3 ) * ( 13 choose 1 )
/ ( 52 choose 13 ).
1 "choose" r = ( 1 ) ** r
2 "choose" r = ( ( 1 ) ** r ) * ( r + 1 )
/ x + 1 \ / x \ / x \
  =   +  
\ r / \ r  1 / \ r /
Newton's Binomial Formula
For any number a (integer) and all values 1 < t < 1, ( 1 + t ) ** a
/ a \ / a \ / a \
= 1 +   * t +   * t ** 2 +   * t ** 3 + ....
\ 1 / \ 2 / \ 3 /
If a is a positive integer, all terms to the right containing powers
higher than t ** a vanish automatically and the formulat is correct
for all t. If a is not a positive integer, the right side represents
an infinite series.
Geometric Series ( 1 < t < 1 )
( 1 + t ) ** 1 = 1  t + t ** 2  t ** 3 + t ** 4  t ** 5 + ....
Taylor expansion of the natural logarithm ( 1 < t < 1 )
(integral of the geometric series)
log ( 1 + t ) = t  ( t ** 2 ) / 2 + ( t ** 3 ) / 3 + ....
If 1 < t < 1, then
log ( 1 / ( 1  t ) ) = t + ( t ** 2 ) / 2 + ( t ** 3 ) / 3 + ....
Stirling's formula
n! ~= ( ( 2 * Pi ) ** 0.5 ) * ( n ** ( n + 0.5 ) ) * exp ( n )
where the sign ~= is used to indicate that the ratio of the two sides
tends to unity as n approaches infinity.
log ( n! ) = log ( 1 ) + log ( 2 ) + ... + log ( n )

Hoppensteadt, F. C. An Introduction to the Mathematics of Neurons.
1986. Chapter 1.

VCON == VoltageControlled Oscillator Neuron
PLL == PhaseLocked Loop
Ohm's Law: V = I * R
Inductors. These are coils of wire wrapped around a metal core.
Current through the coil induces a magnetic field in the core that
creates a voltage.
V = L * I'
The constant L is called the inductance, and it is measured in units
of henrys.
Capacitors. A capacitor is a device that accumulates charge on
plates separated by a nonconductor. The constant C is called the
capacitance, and it is measured in units of farads. In most of the
circuits used here the appropriate units are microfarads
(1.0e6 farads).
Electromotive force. A power supply, like a battery or an
alternating voltage, applies an electromotive force (voltage) to a
circuit. We denote an electromotive force by E.
RLCcircuits are linear.
Kirchhoff's Laws
1. The total voltage measured around any closed loop that can be
drawn in the circuit is zero.
2. The total current into any circuit node is zero.
The "isoclines" on a plot of I vs. V are where both V' = 0 and I' = 0.
Solutions in closed form can only be found for linear circuits.
Harmonic Oscillator of an LC circuit: L * C * V'' + V = E
with natural frequency ( 1 / ( L * C ) ) ** 0.5
LCcircuits, RLCcircuits with no resistance, behave like timers.
Filters are important since they sort out, and allow to pass only
certain frequencies. Their purpose is to eliminate noise from a
signal or to restrict a signal to a size that meets tolerances of
circuit elements farther down stream.
V_In */\/\/\/** V_Out
R 

LowPass 
Filter  C



 Ground

 
V_In * ** V_Out
  

HighPass \
Filter /
\ R
/
\



 Ground


Hoppensteadt, F. C. An Intoduction to the Mathematics of Neurons.
1986. Chapter 8.

Radioactive Decay
Given
x' = alpha * x  decay rate is proportional to its mass)
x ( Time => 0 ) = A  original mass at time 0
dx / x = alpha * dt
ln ( x / A ) = alpha * t
x ( t ) = A * exp ( alpha * t )
Harmonic Oscillator
x'' + ( omega ** 2 ) * x = 0
x ( 0 ) = A
x' ( 0 ) = B
Omega is a known constant called the free frequency.
Given any two linearly independent solutions, say x1 ( t ) and
x2 ( t ), _any_ solution of the harmonic oscillator can be written
as a linear combination of them. That is, the form
x ( t ) = a1 * x1 ( t ) + a2 * x2 ( t ),
where a1 and a2 are free constants, covers all possible solutions.
Pi / 2 radians = 90 degrees
Externally Forced Harmonic Oscillator
x'' + ( omega ** 2 ) * x = f ( t )
where f ( t ) == external forcing function

Mead, Carver. Analog VLSI and Neural Systems. 1989.

epsilon_0 = 8.85e12 farads/meter. When epsilon is stated in terms of
epsilon_0, it is called the dielectric constant.
Force of gravity f_g = G * m1 * m2 / ( r ** 2 )
The force of gravity on Earth (weight ) is
f_g = m * ( M * G / ( r ** 2 ) ) = m * g.
The gravitational field due to the mass M (force per unit mass) is
g = M * G / ( r ** 2 ).
For values of height (h) much smaller than r, on Earth, the
potential energy PE == V = m * g * h.
The gravitational potential, g * h, is the energy per unit mass of
matter at height h above the Earth's surface.
The raise the potential of 1 coulomb of charge by 1 volt requires
1 joule of energy.
The electron volt (e_V) is the energy required to raise the potential
of one electron of charge by 1 volt: 1 e_V = 1.6e19 joules.
The unit of capacitance, coulombs per volt (C/V), is called the farad.
The unit of resistance R, volts per ampere, is called the ohm (Omega).
Q = C * V
I = dQ / dt
I = C * dV / dt
V = I * R ==> R = V / I
mho = siemens = G = 1 / R = I / V
Devices with gain are called active devices.
Newton's Law: Force = mass * acceleration == F = m * a
The distance s traveled in time t by a particle starting from rest
with acceleration a is s = 0.5 * a * ( t ** 2 ).
delta h = 0.5 * a * ( t_f ** 2 ) = ( f / ( 2 ** m ) ) * ( t_f ** 2 ).
The average drift velocity (v_drift) of a large collection of particles
subject to the force f per particle is just the net change in position
delta h per average time t_f between collisions:
v_drift = delta h / t_f = f * t_f / ( 2 * m ).
The force on each particle with charge q in the presence of an electric
field E is f = q * E.
The drift velocity in the electric field E is
v_drift = f * t_f / ( 2 * m ) = q * t_f * E / ( 2 * m ) = mu * E
where the constant mu = q * t_f / ( 2 * m ) is called the
mobility of the particle.
There is a diffusion of particles from regions of higher density to
lower density.
The flow rate (J), given in particles per unit area per second,
can be viewed as a movement of all particles to the right with some
effective diffusion velocity (v_diff): J = N * v_diff.
For electrically charged particles, J usually is given in terms of
charge per second per unit area (current per unit area), and is
called the current density. J = ( N * q ) * v_diff.
In a onedimensional model, a particle in thermal equilibrium has
a mean kinetic energy that defines its temperature (T):
m * v_Tý / 2 = k * T / 2. Here k is the Boltzmann's constant, m is
the mass of the particle, and v_T is called the thermal velocity.
At room temperature, k * T = 0.025 electron volt.
v_diff = ( 1 / ( 2 * N ) ) * ( dN / dh ) * k * T * t_f / m
= D * ( 1 / N ) * ( dN / dh )
The quantity D = k * T * t_f / ( 2 * m ) is called the diffusion
constant of the particle.
The mobility and diffusion constants are related:
D = ( k * T / q ) * mu. This result is called the Einstein relation;
it was discovered by Einstein during his study of Brownian motion.
It reminds us that drift and diffusion are not separate processes,
but rather are two aspects of the behavior of an ensemble of particles
dominated by random thermal motion.
If the temperature were reduced to absolute zero, the entire
atmosphere would condense into a solid sheet about 5 meters thick.
The density of molecules per unit volume in the atmosphere decreases
exponentially with altitude above the earths surface (w is weight):
N = N_0 * exp (  w * h / ( k * T ) ).
For charged particles, the potential energy is q * V.
The voltage V developed in response to a gradient in the concentration
of a charged species, and exhibiting the logarithmic dependence on
concentration shown below, is called the Nernst potential.
V = ( k * T / q ) * ln ( N / N_0 )
In the electrochemistry and biology literature, k * T / q is written
R * T / F.
The Boltzmann distribution describes the exponential decrease in
density of particles in thermal equilibrium with a potential gradient.
N = N_0 * exp ( q * V / ( k * T ) )

Mead, Carver. Analog VLSI and Neural Systems. 1989.
Chapter 3: Transistor Physics

The active devices in electronic systems are called transistors.
Their function is to control the flow of current from one node based
on the potential at another node.
For the planet Earth, the gravitational attraction, temperature, and
molecular weight are such that the atmospheric density decreases by
a factor of e for each approximately 20 km increase in elevation.
The total number of electrons in orbit around an atom of a particular
element is called the atomic number of that element.
Simplified Periodic Table of the Elements
I II III IV V VI VII Zero

H He
Li Be B C N O F Ne
Na Mg Al Si P S Cl Ar
K Zn Ga Ge As Se Br Kr
Rb Cd In Sn Sb Te I Xe
The Group IV elements form a diamond lattice.
Silicon is by far the most commonly used semiconductor.
Boron, aluminum, and gallium are acceptor impurities in silicon;
phosphorous are arsenic are donors (of electrons).
Group III elements can combine with Group V elements to form
diamondlike crystals in which alternate lattice sites are occupied
by atoms of each element. The best known of thes Group IIIV
semiconductors is gallium arsenide, which is used for microwave
transistors and lightemitting diodes.
Group IIVI crystals also are semiconductors. Zinc sulfide is a
common phosphor in television display tubes, and cadmium sulfide was
the earliest widely used photosensitive material.
Group 0 elements such helium, neon, and argon are inert gases.
Covalent bonds are based on sharing electrons such that all elements
can have a full shell, even though some of the electrons taht fill
the outer shell are shared with neighbors. Small communal aggregates
of this sort are called molecules.
Three elements in Group IV of the periodic table crystallize naturally
into a remarkable structure called the diamond lattice: carbon,
silicon, and germanium. Each atom is covalently bonded to four
neighbors arranged at the corners of a regular tetrahedron.
A pure crystal formed from Group IV atoms is called an intrinsic
semiconductor; it is an electrical insulator, because there are no
charged particles free to move around and to carry current.
If we alter the crystal by replacing a small fraction of its atoms
with impurity atoms of Group V, the crystal becomes conductive.
The addition of impurities is called doping.
Group V doping atoms are called donors, because they donate a free
electron to the crystal. The free electrons are negative, and a
semiconductor crystal doped with donors is said to be ntype.
When an electron leaves its donor, the donor is said to be ionized.
An ionized donor has a positive charge because it has lost one
electron.
Group III dopants are called acceptors. The absence of one
electron in a bond is called a hole. Doping a semiconductor with
acceptors renders it conductive, the current being carried by
positive holes. Such a crystal is called ptype.
Heavily doped ntype material is called n+, and heavily doped ptype
material is called p+. The density of impurity atoms is always
small compared to the approximately 5e+22 atoms per cubic centimeter
in the crystal itself.
Because electrons and holes are both charged, and are both used to
carry current, we refer to them generically as charge carriers.
SiO2 == Silicon Oxide == Quartz is an excellent electrical insulator.
Current flows from the source to drain in the region just under the
gate oxide called the channel. This structure was first described
by a freelance inventor named Lilienfeld in a patent issued in 1933.
MOS transistor: Metallic gate, Oxide insulator, Semiconductor channel
polycrystalline silicon == polysilicon == poly
The Fermi level measures the energy of the charge carrier.
The general form of the MOS transistor current is
I = I_0 * exp ( q * Vg / ( k * T ) )
* ( exp ( q * Vs / ( k * T )  exp ( q * Vd / ( k * T ) ) ).
For a transistor with its source connected to the power supply rail,
Vs is equal to zero and
I = I_0 * exp ( q * Vgs / ( k * T ) )
* ( 1  exp ( q * Vds / ( k * T ) ) )
where Vgs and Vds are the gatetosource and drainsourcevoltages.
Because there are charge carriers with positive as well as negative
charge, there are two kinds of MOS transistor: Those using electrons
as their charge carriers are called nchannel, whereas those using
holes are called pchannel; the technology is thus called
complementary MOS, or CMOS.
For positive q (pchannel device), the current increases as the gate
voltage is made negative with respect to the source; for negative q
(nchannel device), the opposite occurs.
k * T is the thermal energy per charge carrier, so the quantity
k * T / q has the units of potential; it is called the
thermal voltage, and its magnitude is 25 mV at room temperature.
A carrier must slide down a potential barrier of k * T / q to raise
its energy by k * T.
Drain Source
 
 
__ __
   
Gate   nChannel Gate o  pChannel
 __ transistor  __ transistor
   
 
 
Source Drain
We put a bubble on the gate of the pchannel symbol to remind us that
the transistor turns on as we make the gate more negative relative to
the source. We normally will draw the positive supply at the top of
the diagram, and the most negative supply at the bottom. For this
reason, the sources of pchannel devices usually are located at the
top, whereas those of nchannel devices normally are at the bottom.
For a given gate voltage, the drain current increases with Vds and
then saturates after a few k * T / q. The current in the flat part
of the curves is nearly independent of Vds and is called the
saturation current, I_sat.

Serway, Raymond A. Physics for Scientists & Engineers, 3/e, Volume 2,
Updated Printing. 1992.

The naturally occurring stone magnetite (Fe3O4) is attracted to iron.
"Elektron" is the Greek word for amber.
The coulomb is the unit of electric charge.
Electric charge exists in discrete packets: q = N * e.
The force between charges varies as the inverse square of their
separation: F ~= 1 / ( r ** 2 ).
Charging a body by induction requires no contact with the body inducing
the charge, unlike conduction.
Coulomb's Law
The magnitude of electric force between two charges is
F = k * abs q1 * abs q2 / ( r ** 2 )
where k is the Coulomb constant.
Current: 1 ampere (A) = 1 coulomb (C) / second.
The Coulomb constant k ~= 9.0e+9 N * ( m ** 2 ) / ( C ** 2 ).
The Coulomb constant k = 1 / ( 4 * Pi * epsilon_0 ) where
epsilon_0 is the permittivity of free space.
The permittivity of free space
epsilon_0 = 8.8542e12 ( C ** 2 ) / ( N * ( m ** 2 ) ).
The smallest unit of charge known in nature is the charge on an
electron or proton with a magnitude abs e = 1.60219e19 C.
1 C = 1 / e = 6.3e+18 electrons (1e6 C ~= charged glass rod).
A metal atom, such as copper, contains one or more outer electrons,
which are weakly bound to the nucleus. When many atoms combine to
form a metal, the socalled free electrons are these outer electrons,
which are not bound to any one atom. These electrons move about the
metal in a manner similar to gas molecules moving in a container.
Table 23.1 Charge and Mass of the Electron, Proton, and Neutron
Particle Charge (C) Mass (kg)
  
Electron (e) 1.6021917e19 9.1095e31
Proton (p) +1.6021917e19 1.67261e27
Neutron (n) 0.0 1.67492e27
Force is a vector quantity with magnitude and direction (not a scalar).
Coulomb's law applies exactly only to point charges or particles.