Index
Note: Page numbers followed by “f” denote figures; “t” tables.
A
Absorbing barriers, 3–5
Absorbing chains
average chain length, 11
conditional expectation, 8
conditional probability, 8
destitute crabs, 10–11
disjoint events, 9
first return probability, 7
geometric distribution, 7–8
Markov chain study, 8
multiplier effect, 11–12
probability of eventual absorption, 8
social mobility model, 10
Absorbing state, 6
Adams, John Quincy, 29–30
Adenomas, 82
Advective equation, 190
Algae growth, 160
Algal patches
bloom patches, 195
concentration, 193
using diffusion equation, 193
patch boundary effect, 194–195
red tides, 192
relationship, 194
reproductive ability, 195
American gray squirrel, 197
Animal coat markings, 210
Apportionment, 30
Hamilton’s proposal, 27–28
inequality, 29
mathematical convenience, 30
minimizing sum of squared differences, 31
recreation schedules, 30
rich history, 26–27
rounding up, 29–30
Webster’s method, 29–30
Arbitrage process, 131
Arbitration rule, 33
Asymptotic stability, 243–244
cyclic attractors, 147–147
point attractor, 162
Aureococcus anophagefferens (A.a.), 171
excitability, 172–174
grazing rate, 171
growth rate, 171
nullclines, 173f
phytoplankton, 171
predation form, 172
quantity, 172
sole equilibrium, 172
Autocatalysis, 177–178
feedback loop for, 178f
model equations, 185
Awkward turns, 72–73
B
BA, See Batting average
Basin of attraction, 147
independent random samples, 90
JS estimator, 91–93
normal distribution, 93
paradoxical, 90
shrinking factor, 93
Bayes’ factor (BF), 82
aspects, 79
coin-tossing, 81
colorectal screening, 78
conditional probability, 80
conditional statements, 77
defendant’s fallacy, 85–86
disjoint events, 80
DNA samples match, 78
fund managers, 79–80
mathematical details, 94–95
odds formula, 85
odds ratio, 81
odds transformation, 85
product rule, 81
sample space, 81
Simpson’s defense, 84–85
Bayesian thinking, 89–90
Ben Franklin’s aphorism, 132
Bernoulli process, 120
Bessel’s equation, 194
BF, See Bayes’; factor
Bifurcation point, 243
Binomial distribution, 115
Bipartite graph, 66
biochemical interactions, 176
biochemical reactions, 177f
blood proteins, 180
double loop, 183
using equations, 178
fatal thrombosis, 177
formation, 179
fourth-order characteristic polynomial, 184f
matrix eigenvalues, 181
Michaelis–Menten equation, 178–179
nullcline approach, 179
oral blood thinning drug, 181
protein Z, 179
Riccati equation, 184–185
self-enhancing loop, 176–177
tissue factor, 176
Blood coagulation, 159
Boom-and-bust scenario, 149–150
Bürger’s equation, 206
Burke’s Theorem, 62
C
Cancerous polyps, See Adenomas
Catastrophe, 126
Chaotic dynamics/randomness
allure of chaos, 169
attractors for biological systems, 168
chaotic attractor, 165–166
fluctuations, 166
mimic actual measles events, 165
numerical results, 167
plausible caricature, 167–168
Chemotaxis, 215
Chi-squared test (χ2 test), 110
baseball examples, 118
Bernoulli variables, 117
chi-squared distribution, 117
empirical distribution, 117
hypothetical distribution, 116–117
null hypothesis, 117
zero mean and unit variance, 117
Childhood diseases, 159
Chromatic number, 74–75
Clean streets, 63
deadheading, 63
household refuse collection, 63
routes, 63
Clustering, 111
Coenobita clypeatus (C. clypeatus), 3
Coincidences, 101
assumptions, 103
birth and death dates, 102–103
birthday problem, 104
lottery example, 102
mathematical details, 106–108
near-coincidence, 104
Poisson distribution, 103
probability calculation, 101–102
sample spaces in probability, 101
source, 102
winning draw, 104
Colorability, 67
Colorectal screening, 78
adenomas, 82
Bayes’ rule, 84
colonoscopy effectiveness, 82
contradictions, 82
fallacy, 83
goal, 82
hemoccult test, 84
sigmoidoscopy, 83
upper colon lesions, 83
Commercial fishing, 139
Competitive equilibrium, 154–155
Competitive exclusion, 155
Conditional probability, 80
Conservation of mass, 189–192
Contested-garment principle, 33
Crabs and criminals
C. clypeatus, 3
country home, 1
data-gathering effort, 2
larger-scale problems in sociology, 2
mathematical model of movement, 2
queuing, 3
recidivist, 2
social ladder, 1–2
social mobility, 2
States, 2
D
Deadheading, 63
Degree of a node, 64–65
Delbrück–Luria experiment, 116
Dictyostelium discoideum (D. discoideum), 210
advective equation, 190
conservation-of-mass argument, 190–191
diffusion-driven instability, 209–210
external sources, 189
Fick’s law, 188
molecular, 188
Newton’s law, 188
quantity calculation, 189
source function calculation, 191–192
Taylor’s theorem, 189
two-dimensional flow, 192
coin-tossing experiment, 112
Poisson process, 112
theoretical and actual data comparison, 111–112
Discount factor, 152
loopy DNA, 221–224
Drug–placebo experiment, 86
E
Edge-routing problems, 72
Edges, 74
80–20 rule, 129–130
El Niño, 139
Empirical verification, 49
ESP, See Extrasensory perception
Euclidean metric, 47
Euler tours, 63–66
bipartite graph, 66
colorability, 67
degree of node, 64–65
graph, 63–64
Konigsberg bridges, 63–64
odd-degree nodes, 65
odd-order cycle, 67
Excitability, 172–174
Extrasensory perception (ESP), 88–89
Extreme-tail events, 95
Eyeballing, 73
F
Fade-out condition, 164–165
False positives, 77–78
Fire companies, deployment of
cost saving measures, 60
high-density area, 59
homogeneous sectors, 58
integer program, 58–59
often-conflicting interests, 59
repositioning, illustration of, 60–61
response neighborhoods, 60
service times, 58
short-term advantage, 60–61
Fire companies operation, 43
Fire companies, optimal deployment of
cost saving measures, 60
high-density area, 59
homogeneous sectors, 58
integer program, 58–59
often-conflicting interests, 59
repositioning, illustration of, 60–61
response neighborhoods, 60
service times, 58
short-term advantage, 60–61
Fire companies role
See also Public services
See also Public services
conflagration, 53
equilibrium hypothesis, 54
equilibrium probability, 56–57
firefighting units delay, 55
multiple-dispatch strategy, 58
Poisson approximation, 55
Poisson processes in tandem, 55
single-stage fires, 53
transient effects, 53–54
transition diagram, 56
two-stage system, 56
Fire safety and precautions
alarm rates, 40
engine company, 39
fairness criteria, 40
firefighting units, 40
geographic distribution, 39
ladder company, 39
primary mission of, 39
Firefighting units, 40
Fishery model
See also Restricted-access fishery
See also Restricted-access fishery
anchovies and sardines, 145
bifurcation point, 144
boom-and-bust scenario, 149–150
cost-to-price ratio, 145
using curves, 142
differential equation, 140–141
equilibria coalesce, 144
fish harvesting model equations, 150
fish stocks, 144
hysteresis effect, 144
interpretation, 141
intersection, 143
nonlinear fishery equations, 152
operating expenses, 140–141
Peruvian anchovy fishery, 145
recruitment failures, 144–145
sardines and species, 140
scalar equation, 141–142
stable cyclic trajectory, 151
stable equilibrium values, 145f
Fisher’s equation, 196
G
Game theory, 156
Gauss’s linking formula, 229
Geometric distribution, 8
Geometric series, 52
Graph, 63–64
Gray squirrel spread
American gray squirrel, 197
components, 202–203
critical minimum speed calculations, 203
derivations, 200–201
diffusion and spatial dependence, 199
invading squirrels, 197–198
numerical simulation, 198–199
population densities, 198
second-order equation, 201
in space, 198f
wavelike function, 200f
H
Hamilton, Alexander, 27–28
Hamilton’s proposal, 27–28
Harvesting effort, 152
Hitch, 21
Hopf bifurcation theorem, 147–149
Hypergeometric distribution, 107
I
Inequality, 36
Infectious disease, 160
Inheritance rule, 34
Integer program, 22
Integer-programming problem, 71
Inverse square-root law
average number of units, 48
convex function, 48
density function, 45–47
empirical verification, 49
Euclidean metric, 47
fire incidents, 45
Poisson-distributed fire companies, 45
right-angle metric, 45
spatial Poisson distribution, 44–45
J
Jacobean matrix, 148
James–Stein approximation (JS approximation), 90
Jefferson, Thomas, 28
K
Konigsberg bridges, 63–65
Konigsberg puzzle, 63–64
L
Linking number, 227–229
Logistic equation, 196
Logistic growth law, 152
Loopy DNA
computations, 223–224
cross-sectional torus slice, 222
helical curve winding, 222f
plane slicing torus, 223f
ribbon model with different values, 224f
unit vector calculation equations, 223
M
Manpower scheduling
adding equations, 24
admissible recreation schedules, 23
advantageous weekend break, 24
available workers and demand mismatch, 22–23
basic argument, 21
compactly in matrix notation, 20
compromise solution, 24–26
constraining relations, 20
existing rotating schedule change, 23
hitch, 21
integer program, 22
integer programming formulation, 26
recreation choice, 21–22
recreation schedules, 19
rotating schedule, 20–21
scheduling workers in perspective, 19
trade-off, 24–26
Market volatility
Bear-Stearns near-collapse, 133–134
mean variance analysis, 134
power laws and fat tails, 133
self-enforcing process, 134
Markov chain, 4
Maximum sustainable yield, 143
Measles epidemics
actual data on, 163
categories, 160
childhood epidemics, 160
childhood illnesses, 160–161
coupled equations, 162
disease contact rate, 161
disease pathogen, 163
fade-out condition, 164–165
infectives, 161
population, 161
threshold condition, 165
Melanocytes, 211
Memoryless property, 42–43
Michaelis–Menten equation, 178–179
Molecular diffusion, 188
Morphogens, 209
Multiple-dispatch strategy, 58
Multiplicative effect, 132
Multiplier effect, 11–12
N
Nash competitive strategy, 156
Near-coincidence, 104
Nodes, 74
Nonzero polarity, 69–70
Normal distribution, 93
Null hypothesis, 86
Nullcline approach, 179
O
Odd-order cycle, 67
One-percenters, 129
80–20 rule, 129–130
conditional probability, 129
household incomes, 129
income distribution, 128–129
pile, 131
power law distributions, 128
probability density, 129
self-organized criticality, 131
top-heavy power function, 130
word fraction, 130
Zipf plot, 130
Oral blood thinning drug, 181
Orbit, 165–166
P
Pagarus longicarpus (P. longicarpus), 6
Pareto distribution, 132
Pari delicto principle, 34–35
Pattern formation in biology
amoeba aggregation onset, 211f
intuitive demonstration, 210f
mathematical models, 210–211
morphogens, 209
Turing’s model, 209–210
Turing’s reaction–diffusion model, 210
Peruvian anchovy fishery, 145
Phytoplankton, 171
Poisson approximation, 108
BA, 113
cluster, 109–110
fire alarm response times, 110
probabilities, 109
Poisson assumption, 113
assumptions, 111
coin-tossing experiment, 111
fire alarms, 108
fire department operations, 110
random processes, characteristic, 109
Poisson events
conditional probability, 42
fire companies operation, 43
independent increments, 40–41
independent Poisson processes, 42–43
nonnegative and integer-valued random, 40
peak alarm period, 43
probability distribution of random variable, 41
queuing theory, 44
service time, 43
spatial counting process, 44
stationary increments, 41
at random time, 42
spatial processes, 44
temporal processes, 113
Political districting, 37
Ponzi scheme, 34
Power laws, 125
asymptotic relationship, 126
distribution, 125
energy in wind turbulence, 127
fat tails, 126
first-order differential equation, 127
infinite values, 128
linear relationship, 127
log-log scale, 127
log-normal distribution, 128
mathematical details, 135–136
scale invariant, 127
scaling, 127–128
Predator-mediated coexistence, 169
model equations, 169
single attracting equilibrium, 169–171
two pray, one predator model, 170f
Prosecutor’s fallacy, 78
Public services
apportionment problem, 18
fair allocation of scarce resources, 18–19
gerrymandering, 18
mathematical details, 35–36
mathematical framework, 17
nonpartisan mathematical approach, 19
representatives and direct taxes, 18
trade-off, 17
work requirements, 17
worker assignment, 17
Q
Queuing theory, 44
R
Rabid foxes
Bürger’s equation, 206
differential equation, 206
epidemic model, 203
flow equation, 205
marginal connection, 205
planar system, 204
shock wave, 207
spatial spread, 203–204
steep wave front, 207f
traffic equation, 204f
traveling-wave solutions, 204
Random mutations, 114–115
Random string, 118
Reaction, 188
advective equation, 190
conservation of mass, 189–191
external sources, 189
flow of particles, 191
integral sign, 190
quantity calculation, 189
source function calculation, 191–192
Taylor’s theorem, 189
two-dimensional flow, 192
Recidivism
absorbing, 12
in effect, 13
Markov chain model conditions, 13–14
one-step transition matrix, 13
probability of recidivism, 14
recidivate, 12
recidivism different estimates, 14
small effect, 14
transitions between states, 12
Recreation schedules, 19
Recruitment failures, 144–145
Repeller, 149
Restricted-access fishery
competitive equilibrium, 154–155
curves intersection, 153–154
equilibrium, 152
fish density and harvesting effort, 152
harvesting policy, 152
logistic growth law, 152
maxima function, 155
unstable equilbria and cyclic behavior, 146–149
Riccati equation, 184–185
Right-angle metric, 45
S
Scale-free property, 127
Scaling, 127–128
Sciurus carolinensis, 187–188
Sciurus vulgaris, 187–188
Self-enforcing process, 134
Self-organized criticality, 131
Shock wave, 207
Sigmoidoscopy, 83
Simplifying assumption, 51
Simpson’s paradox, 97–98
invasive treatment, 97
kidney stones, 97
vector space approach, 97–98
Skeptical Bayesians, 86–90
Bayes’ formula, 87–88
Bayesian thinking, 89–90
ESP, 88–89
garden-variety statistical technique, 87
hypothesis testing, 86
level of significance, 87
null hypothesis, 86–87
placebo effect, 88
posterior probability, 88
statistic, 86–87
Slime molds
diffusive flux with diffusivity, 215
instability, threshold condition for, 217
oscillatory solution, 216
single-cell amoeba, 215
Turing equations, 215–216
Slump, 111
Social mobility, 2
absorbing state, 6
model, 10
P. longicarpus, 6
Space curves, 219
Gauss linking number, 225–229
link and twist, 221
writhe, 219–221
Spatial Poisson approximation, 109
Spatial Poisson process, 44
Splotches, See Stripes and splotches
Standard trigonometric formula, 49
Stationary increments, 41
Steinbeck, John, 139
Stein’s paradox, 90
Stein–James paradox, 96–97
Stokes’ theorem, 227
Street sweeping
awkward turns, 72–73
directed networks, 68
directed-graph network representation, 69f
edge-routing problems, 72
household refuse collection, 68
integer-programming problem, 71
with mechanical brooms, 67
negative and positive polarity, 70
one-way and two-way street, 68f
optimization program, 69
optimum values, 71
polarity of node, 69–70
streets flushing, 67–68
subset of edges, 69f
supplementary links, 71f
total demand and supply, 70
total sweep time, 72
truck routes, 72
urban street network, 68–69
Stripes and splotches
boundary conditions, 213
conical surface, 212
critical threshold, 214–215
independent variables, 212
using linear theory, 215
melanocytes, 211
no-flux boundary conditions, 213
nonlinear reaction–diffusion equations, 211
stripes and splotches formation, 212f
time-independent solution, 213
Turing’s reaction–diffusion hypothesis, 214
Superhelix, 224
Susceptibles, 160–161
T
Talmud and Madoff’s scheme, 31
arbitration rule, 33
claw back, 34–35
contested-garment principle, 33
contested-garment problem, 32
fairness criterion, 32–33
finding corresponding division, 32
inheritance rule, 34
pari delicto principle, 34–35
Ponzi scheme, 34
puzzling division, 31
rationale, 31–32
two-person contested garment rule, 33
Temporal Poisson process, 113
TF, See Tissue factor
Tissue factor (TF), 176
Tour graph, 74–75
Traffic congestion equation, 204f
Transition matrix, 3
Transitions between states
absorbing states, 4
nonabsorbing states, 4
random walk with absorbing barriers, 3
state of system, 4
transition matrix, 3–5
Traveling waves, 196
Traveling-wave function
comments, 196
for different values, 196f
using equation, 196
solution, 196
temporal evolution, 197f
Twisting, 220
Two-stage system, 55
U
Urban grid, encumbrance of, 49–50
V
Vehicle capacity, 73
Vehicle routing, 76
Vehicle scheduling
daily and weekly schedules, 74
eyeballing, 73
feasible truck schedule, 73
graph chromatic number, 74
independent sets, 75
red or green tours, 74
schedules selection, 73–74
service-frequency requirements, 74–75
six-times-a-week sites, 74
Viral contamination
See also Blood clotting
See also Blood clotting
algae cells, 174–175
algebraic juggling, 175
cubic characteristic polynomial, 176
model equations, 175f
non-negative solution, 175–176
virus-free equilibrium, 176
W
Webster, Daniel, 29–30
Webster’s method, 29–30
Word fraction, 130
Writhe, 221
Z
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