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Chapters
Introduction to Mathematical Modeling

Introduction to Mathematical Modeling

Terminology
Simulation Pipeline
Examples of Applications
Derivation of Models
Analysis of Models
Classification of Models
Scales, Hierarchy

← PreviousPartial Differential Equations

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Index

A

abstract model
accuracy
Adams–Moulton method
admissible schedule
admissible schedule (job-shop)
alternative-caution strategy (min-max regret)
analytic solution
approximate-numerical method
arbitrary machines
arc length
arms race of two superpowers
arrival process
arrival rate
Arrow's impossibility theorem
asymmetric (relation)
attractive equilibrium
augmented precedence graph

B

backward recurrence time
best response function
bijective function
binary relation
boundary
boundary point
boundary value problem (BVP)
bounded set

C

D

E

eigenvalue and eigenvector
eigenvalues of a diagonal matrix
elementary queuing system
elliptic PDE
embed-test definition
embed-test theorem
equilibrium point
Euclidean norm
Euler description
Euler method
event rate
expected payoff (mixed strategies)
expected value
explicit method
external dictator

F

feedback control
filling
finite difference method
finite element method
floating-point number
forward recurrence time
free-flow phase
Fubini's theorem
full-risk strategy (max-max payoff)
function of a random variable
functional unit
fundamental diagram
fundamental relationship (traffic flow)
fundamental theorem of calculus
fuzzification
fuzzy complement
fuzzy control system
fuzzy inference
fuzzy intersection
fuzzy rule base
fuzzy set
fuzzy union

G

Gauss's divergence theorem
generic decision model
geometric distribution
global discretization error
global extremum
gradient
growth rate

H

harmonic number
hazard rate
heat equation
Hessian matrix
Heun method
heuristic
homogeneous birth–death process
homogeneous Markov process
hyperbolic PDE

I

ideal arithmetic
identical machines
identity matrix
ill-posed problem
image (trace) of a curve
implicit Euler method
implicit method
independence of irrelevant alternatives
independent process
initial condition
initial value problem (IVP)
injective function
inner point
inner product
inner product (geometric form)
instationary (PDE)
integration by parts
integration rules
interior
internal dictator
interpolation polynomial (multistep)
inverse problem
inverse transform sampling
irreducible chain
isoline
iteration matrix

J

Jacobian determinant
Jacobian matrix
job
job-shop model

K

k-times continuously partially differentiable
Kendall notation
Kolmogorov–Smirnov test

L

Lagrange description
Laplace equation
Laplace operator
lead time
linear 1st-order PDE (in 2D)
linear birth and death rates
linear congruential generator
linear feedback control
linear PDE
linear PDE of 2nd order
linguistic variable
Little's law
load
local convergence of Newton's method
local discretization error
local extremum
logistic growth model

M

majority decision
makespan
Malthus model
Markov process
mathematical modeling
maximum throughput
memorylessness
method of steepest descent
midpoint rule
minimax theorem
mixed strategy
model
model hierarchy
multigrid method
multiscale problem
multistep method
multivariate function

N

nabla operator
negative exponential distribution
Neumann boundary condition (ODE)
Neumann boundary condition (PDE)
Newton iteration (multivariate)
no-dictator condition
non-feedback control
normal distribution
null hypothesis
nullcline
number of visits

O

one-step method
open queuing network
open set
order (of a method)
ordinary differential equation (ODE)

P

Q

q-th moment
quadratic formula
qualitatively identical machines
quasi-order
quasi-stationary (traffic flow)
quasilinear 1st-order PDE (in 2D)
queuing network
queuing system

R

random variable
random walk
rank addition
rank mapping
recurrence probability
recurrent state
reduction to a first-order system
reflexive (relation)
regret
regular surface
relaxation method
remaining time
repulsive equilibrium
residual
resolution (floating-point)
restriction
root
rotation
round-off error
rounding
Runge–Kutta method (RK4)
runs test

S

T

tangent vectors
task stream
Taylor polynomial
theorem of underestimation (Fulkerson 1962)
throughput
topological sort
trace
trace from eigenvalues
traffic bottleneck
traffic density
traffic equation
traffic flow
traffic velocity
trajectory
transition probability
transition rate
transitive (relation)
translation invariance (differentiation)
transpose
trial function
trigonometric derivatives
trigonometric integrals
triple integral (box domain)
triple integral (standard domain)
truncation error
two-player zero-sum game
two-species model

U

uniform distribution
upwind discretization
utilization

V

V-cycle
validation
variance
vector field
vector projection
vector surface integral
Verhulst saturation model
Vieta's formulas
voting paradox

W

well-posed problem

#

1st-order linear PDE with constant coefficients
α-cut
ε-ball

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