Activity in linear programming, 214, 215, 220
unit, 214
Adaptive control process, 204, 206
Addition of convolution quotients, 12
Adjoint of matrix, 322
Aitkin method, 396
Algebra of convolution quotients, 13
Alternating-direction method, 405–407
Alternative symmetrization, 435–437, 439
Amplitude, complex, 285
Anisotropic medium, 301
Annular Hankel transform, 96
Antiderivative of a distribution, 33
Artificial variable, 227
Asymptotic formula, 133, 137, 147–150, 161, 162
Asymptotic series, 133–163
Atomic-interaction phenomena, 133
Attached shock, 371
Axis of symmetry, 425, 428, 429, 439, 440
Banach space, 49
Bang-bang control process, 200, 202
Basic feasible solution, 224, 225
Basic variables in linear programming, 222
Basis, 61
Bernoulli equation, 365
Bernoulli number, 144
Bernoulli theorem, 262
Bessel function, 92, 125, 136–138, 142
Bessel transform, 70
Bilateral network, 299
Bound of operator, 109
Boundary conditions, 117, 119–128
Boundary mesh point, 379
Boundary-value problem, 23, 119–128, 376–384
for bubble, 365–368
eigenvalue for, 429
for oscillating piston, 90
for supersonic flow, 87
two-point, 197
Branching process, 261, 264–270
multidemensional, 270–275
Bromwich contour integral, 54, 55
Brouwer fixed-point theorem, 323
Brownian motion, 169
Bubble problem, 365–368
Calculus of variations, 196, 197
boundary-value problem, 197
computational aspects, 202, 206
constraints, 197
inequality, 200
implicit functionals, 198, 200, 201
quadratic criteria, 198, 199, 203
relative extrema, 197
Canonical form in linear programming, 222
Canonical system of equations, 352
Capacity, of channel, 185, 186, 191–193
Catastrophies, catalogue, 197, 198
Cauchy problem, 101, 104, 111, 112, 347–371
for elliptic equation, 360–365
Cauchy sequence, 106
Cavity-flow problem, 408–410
Center of symmetry, 439–441
Central limit theorem, 167, 172
Channel, 183
binary symmetric, 192
fundamental theorem for, 187
inequality for, 187–190
entropy, 183–185
capacity, 191
indecomposability, 190
probability law, 183
state, 190
attainable, 190
single, 190
Characteristic curves of differential equation, 350
Chebyshev polynomial, 395
Closed operator, 108
Closure principle, 315–317, 319–321
maximum error probability for, 183
Coin-tossing game, 171
Commutative networks, 297
Commutative ring, 10
Complement, orthogonal, 107
Complete Hilbert space, 106
Complex amplitude, 285
Complex euclidean space, 114, 115
Complex Fourier transform (see Fourier transform)
Complex potential, 365
Complex row vector, 322
length, 322
Composite transformation, 272
Conditional entropy, 185
Cone, 24
Connected network, 383
Constant distribution, 33
Constraints in calculus of variations, 197, 198
Continuous operator, 108
Contour integral of Bromwich, 54, 55
Contraction operator, 111–114, 131
Control, terminal, 196
adaptive, 204–206
Control process, bang-bang, 200, 202
deterministic, 196–202
feedback, 195
illustrative example, 206–209
deterministic case, 207
stochastic, 202–204
Convergence, of convolution quotients, 15, 16
Convolution, 10
Convolution field, differential operations in, 13
Convolution multiplication, 10
Convolution quotients, 11
addition, 12
algebra, 13
field, 11–13
inverse element in, 13
of functions on finite interval, 24
multiplication, 12
parametric, 18
Convolution ring, integral operations, 10
Convolution theorem, 79, 80, 88, 97
Coset, 122
Crank-Nicolson difference equation, 403, 404
Crank-Nicolson method, 376, 402–406
CURE program, 412
Current function, 408
Delta distribution, 28
Delta function, 5–8, 79, 102, 171
Demand, 230
Detached-shock problem, 368–371
Deterministic control process, 196–202
Dielectric sheet, 282
Difference equation, 169
Differentiable distribution, 48
Differential equation, in distributions, 45–47
Differential operator, 13
linear, 57
Differentiation, of distribution, 30, 31
Diffusion equation, 23, 85, 128, 167
Diffusion operator, 109
Diffusion problem, 329
Dirac delta function (see Delta function)
Direct product of distributions, 40
Dirichlet integral, 431–437
Discrete distribution, 243
Discrete time, 231
Dissipative network, 292, 295, 322
Dissipative operator, 113, 118
Distilling of oil, 217
Distribution, 26–29
antiderivative, 33
of arbitrary growth, 28
constant, 33
convolution, 40
delta, 28
differentiable, 48
differential equation in, 45–47
rules for, 32
direct product, 40
of finite order, 41
as generalized derivative of integrable functions, 48
geometric, 175
Heaviside, 30
differentiation, 31
norm for, 49n.
null set, 30
order, 41
primitive, 33
repeated, 34
probability, 243
in several variables, 39
of slow growth, 28
support, 30
value on an interval, 29
vector-valued, 48
Divergent series, 134–140
Division in convolution ring, 11
Douglas-Rachford method, 397, 407
Duhamel integral, 10
Duhamel superposition theorem, 79
Duration of busy period, 175–177
expected, 175–178
mean, 175
Dynamic programming, 196, 201, 202, 205
Eigenfunction, 59
normalized, 60
orthogonal, 59
Eigenvalue, 59, 96, 113, 117, 274
first, 429
Eigenvector, 274
Elasticity, 431
Electrostatic capacity, of condenser, 435
of sphere, 435
of tetrahedron, 435
Element, inverse, in field of convolution quotients, 13
Elliptic equation, 87, 360–368
computational methods for, 376, 378, 413–415
Embedding of convolution ring, 11
Energy-dissipative system, 115
Energy invariant, 324
Enthalpy, stagnation, 368
conditional, 185
Equation, diffusion, 23
elliptic (see Elliptic equation)
parabolic (see Parabolic equation)
Equivalent ordered pairs in convolution ring, 11
Error, rounding, 357
truncation, 357
Euler formula, 148
Euler transformation, 145
Euler-Maclaurin sum, 144
Euler-Mascheroni constant, 135
Eulerizing, 144–146
Evaluation of a function, 26
Exogenous flow, 215
Expected duration of busy period, 175–178
continued, 179
Expected value, 273
Exponential Fourier transform (see Fourier transform)
Exponential operator function, 21
Extension of operator, 108–110, 118–128
Extrapolated Liebmann method, 375, 388
Family of characteristic curves, 350
Feasible solution in linear programming, 221, 223
initial, 226
Feedback control, 195
Field of convolution quotients, differential operations in, 13
Finitary process, 191
Finite-difference method, 356–360
Finite-dimensional space, 103, 104, 262
First come, first served rule, 174
Fixed point, 297
Fixed-point theorem, 323
Flash tower in oil refinery, 216
Fokker-Planck equation, 167, 170–174
Force, impulse, 5
instantaneous, 5
Forward-difference method, 376, 397–402
Four-terminal network, 286
Fourier-Bessel transform, 70
Fourier cosine formula, 72
Fourier cosine transform, 70–77, 88, 89
Fourier integral formula, 72
Fourier sine formula, 72
Fourier sine transform, 70–77, 88, 89, 97
Fourier transform, 42–45, 66, 69–73, 77
finite, 92–96
of locally integrable function, 44
pair, 92
of testing function, 43
Fractional integration, 10
Frobenius-Perron theorem, 273, 277, 278
Function, 5
delta, 5–8
evaluation of, 26
as weak limit of ordinary function, 48
improper, 6
impulse, 5
integrable, 11
locally, 25
null, 26
continuous, 18
differentiation, 19
exponential, 21
integration, 20
logarithmic, 21
parametric, 18
perfect, 24
action on, 25
continuous, 26
linear, 26
Functional equation, 205
Game, against Nature, 203, 204
pursuit, 204
Game theory, 202–204
Gamma function, 140, 141, 160–162
Gas flow, 353–355
Gauss elimination method, 374, 385
Gauss-Seidel method, 374, 375, 385–389, 403, 410
Generalized differential equation, 14, 15
Generalized differentiation, 7, 8
Generalized function (see Function)
Generating function of Laplace, 266
Generating transformation, 271
Generator, infinitesimal, 111, 112
Geometric distribution, 175
Gerschgorin error formula, 384
Global lateral condition, 118
Graph of operator, 108
Graph topology, 108
Group, 101
Hadamard criterion, 100, 101, 348, 360
Hankel transform, 66, 70, 75–77, 90–96
annular, 96
finite, 92
Heat conduction, 85–87, 94, 330, 373, 407
Heat equation, 85–87, 97, 101, 104, 112, 167, 378
Heaviside distribution, 30
differentiation, 31
Heaviside operational calculus, 13, 14
Heaviside operational rule, 78
Heaviside shifting theorem, 78, 88
Hilbert space, vii, 104–114, 283, 330–332
complete, 106
operator on, 108–114
subspace, 107
Huygens’ principle, 315
Huygens’ wavelet, 333
Hydrodynamic stability, 345
Hydroelectric generation, 240–242
with stochastic inflow, 246–248
Hydromagnetic stability, 345
Hyperbolic equation, 8, 24, 349–360
Hyperbolic system, 114–119
IBM 704 computer, 412
Ignition temperature, 412
Imbedding, invariant, 283
Importance sampling, 275
Improper function, 6
Impulse function, 5
Impulsive force, 5
Inequality, for channel entropy, 184, 185
of Schwarz, 105, 109, 120, 312, 322
Inequality system, linear, 213
Infinite-dimensional space, 104–110
Infinitesimal generator, 111, 112
Information, mutual, 185
Initial-value problem, 63–65, 88–90, 112
for circular membrane, 123
in fluid mechanics, 347–371
for heat conduction, 85
in Heaviside theory, 53–55, 59
in semigroup theory, 100–102, 115, 129
for wave motion, 82, 89, 94, 95
well-formulated, 100, 101, 348, 360
Inner iteration, 411
Inner product of vectors, 105, 106, 112, 126
Input-output coefficients, 215
Instantaneous force, 5
Integrability condition, 125
Integrable function, 11
Duhamel, 10
Lebesgue, 7
Riemann, 7
Stieltjes, 8
Integral operations in convolution rings, 10, 13
Integral transform, 68
Integration, fractional, 10
of operator function, 20
Interior mesh point, 379
irregular, 381
regular, 379
Invariant, two-sided, 328
Invariant imbedding, 283
Inventory models, 232–236
econometric, 235
history, 233–237
with lagged delivery, 255–257
with random supply, 252–255
stationary, 249
discount rate for, 229
Inventory process, dynamic stochastic, 244–246
factors in, 229
ordering, 229
penalty costs, 229
quadratic costs, 236
revenues, 230
salvage, 230
storage, 229
Inverse of operator, 110
Inverse element in convolution quotient field, 13
Isotropic medium, 305
Item in linear programming, 214, 215, 220
Iteration, inner, 411
outer, 411
Jacobi method, 386
Jacobian, 272
Kendall’s taxicab-stand example, 181
Kolmogoroff-Smirnoff distribution, 243
Kronecker delta, 92
Lag in delivery, 231
Laplace equation, 97, 332, 360
numerical solution, 373, 375, 378–394
Laplace transform, 15, 27, 66, 74–89
of derivatives, 78
of differential equation, 69
semigroup methods for, 112
Laplace-transform pair, 74
Laplace’s method, 147–150
Last come, first served rule, 175
Lateral condition for initial-value problem, 115
global, 118
Law of large numbers, 167
Lebesgue integral, 7
Legendre transform, 96
Liebmann method, 388
Limit of convolution quotients, 15, 16
Line, of symmetrization, 424, 425, 437, 440
Line relaxation, 396
Linear criteria for variational problems, 199
Linear inequality systems, 213
Linear programming, 213–227
unit, 214
complete, 214
algebraic statement, 220–222
artificial variable, 227
basic feasible solution, 224, 225
basic variables, 222
canonical form, 222
central problem, 213
cost in, 215
cycle in, 226
equivalent system, 223
exogenous flow, 215
optimal, 223–225
general iterative procedure, 225, 226
infeasibility form, 227
input-output coefficients, 215
precious, 214
material balance equation, 214, 215
model, 213–215
nonbasic variables, 223
objective function, 221
of oil-refinery problem, 216–220
profit, 215
simplex method, 222
standard form, 222
Linear vector space, 57
Linearity in calculus of variations, 198
Linearity conditions, 102
Linearization in transformation theory, 309, 310
Local boundary conditions, 117
Locally integrable function, 25
Logarithmic operator function, 21
Lossless plane sheet, 284
Lossless transmission line, 287
Management science, 213
Marching process, 398
Markov process, 168, 175, 248, 277
Matched network, 296
Material balance equation, 214, 215
adjoint, 322
Matrix, of coefficients, 351
dissipative, 314
Frobenius-Perron theorem for, 273
identity, 321
Jacobian, 272
norm, 322
determinant, 298
representation of canonical system, 353
scattering (see Scattering matrix)
self-adjoint, 322
star product, 286
transpose, 113
zero, 113
Maxima, geometric, 424
Maximal dissipative operator, 113, 117–131
Maximum error probability, 183
Maxwell’s equations, 305
Mean displacement, 170
Mean duration of busy period, 175
Mean-square displacement, 171
Measuring probe, 288
Medium, anisotropic, 301
isotropic, 305
nonuniform dielectric, 306–308
principal frequency, 420, 422, 432
uniform vibrating, 420
adjacent, 379
boundary, 379
interior, 379
Method of simultaneous displacements, 386
Minima, geometric, 424
Minimal operator, 118
Minimum principle, 429, 430, 433
Model building in linear programming, 213–215
Modern mathematics, 338
Moments, 272
Monte Carlo method, 261
application, atomic, 276
branching process, 264–275
geometric, 262
Monte Carlo method, application, thermodynamic, 276, 277
Multiple Fourier transform, 75
Multiplication, convolution, 10
of convolution quotients, 12
by scalars, 12
Multiplicative process (see Branching process)
Mutual information, 185
Nature, game against, 203, 204
Negative binomial distribution, 243
Network, 283
bilateral, 299
commutative, 297
connected, 383
reflectively, 295
four-terminal, 286
passive, 292
reciprocal, 299
series, 298
shunt, 293
Neutron-flux distribution, 373
Nonbasic variables, 223
Nonsingular transmission, 289
Norm, of matrix, 322
of operator, 109
of vector, 105
Normal distribution, 167, 171, 243
Normalized eigenfunction, 60
Nuclear-reactor problem, 410
Null function, 26
Null set of distribution, 30
Object in transmission theory, 283
Objective function in linear programming, 221
Obstacle, 283
Oil refinery, 216–220
Operational calculus, 9, 13, 14
Operations research, 213
Operator, 13
bound of, 109
closed, 108
continuous, 108
contraction, 111–114
differential, 13
linear, 57
calculus, 52
Heaviside theory, 53–56
operations on, 51
Operator, diffusion, 109
minimal, 118
eigenfunction, 59
graph, 108
on Hilbert space, 110–114
integral, 13
inverse, 110
linear, 57
norm, 109
resolvent, 110
resolvent set, 110
retraction, 130
self-adjoint, 60
shift, 20
spectral representation, 62
Operator function (see Function)
Optimal feasible solution, 223–225
Optimality principle, 245
Ordered pair in convolution ring, 11
equivalent, 11
ORDVAC computer, 413
Orthogonal complement, 107
Orthogonal eigenfunctions, 59
Orthogonal vectors, 107
Oscillating piston, 90
Outer iteration, 411
numerical method for, 397–402, 405–407
semigroup method for, 128
SPADE project for solving, 413–415
Parametric convolution quotients, 18
Parametric operator function, 18
Pascal distribution, 243
Passive network, 292
Passive transmission line, 287
Path in SPADE routine, 415
Peaceman-Rachford method, 375, 390–394, 406, 412
Perfect function, 24
Petit canonical ensemble, 276, 277
Phase I of linear programming, 226, 227
Phase II of linear programming, 226, 227
Piston, oscillating, 90
Plane of symmetrization, 426, 427, 433
Point record, 415
Poisson distribution, 243, 257
Poisson formula, 332
Pólya frequency function, 243, 245
Polynomial in generalized derivatives, 14
Polytropic gas, 355
Power-absorption coefficient, 292
Poynting vector, 117
Primitive of distribution, 33
Principal frequency, 420–423, 428, 432, 433, 437–440
“asymptotic” values, 423
highest, 441
for quadrilateral, 423
for triangle, 423
Principle, of optimality, 201, 245
of shortest path, 426
Probability, 167
application to transmission theory, 312, 313
distribution, 167, 171, 206, 243
Probability law of channel, 183
Probability scattering matrix, 328–330
Probe, measuring, 288
Production scheduling, 237–239
Projection of vector, 107
Properties of symmetrization, 425
Pursuit games, 204
QED code, 411
Quadratic costs, 236
Quadratic criteria, 198
Quadratic transformation, 278
Quaternion, 289
Queue discipline, 174
variance, 181
Quotient, convolution, 11
Random number, 262
in combinatorial problems, 263, 264
generation, 263
in geometric problems, 262
in queue problems, 174
Random variable, 168
in information theory, 184, 185
mutual information in, 185
in theory of control processes, 205
analytic discussion, 340–343
Rayleigh line, 344
Rayleigh problem, 435
Rayleigh ratio, 430–433, 438, 439
Rayleigh-Ritz method, 430
Receiver, 182
Reciprocal network, 299
Refinery flow problem, 216
Reflection coefficient, 283, 319
Regular mesh point, 379
Relaxation factor, 375
Relaxation method, 377
Residue class of cosets, 122
Resolvent of operator, 110
Resolvent set of operator, 110
Retraction of operator, 130
Riabouchinsky model, 408
Riccati equations, 317
Richardson method, 394–396
Riemann integral, 7
Ring, commutative, 10
convolution, 10
Rotational flow, 338–345
Rayleigh criterion for, 339–343
stability, 343
Rounding error, 357
Ruin problem, 167, 173, 174, 178
Saddle point, 152
Scattering matrix, 282, 283, 286, 313–319
probability, 328–330
Schwarz inequality, 105, 109, 120, 312, 322
Self-adjoint matrix, 322
Self-adjoint operator, 60
Semigroup of operators, 101–104, 110–114
Semigroup method, 112–131
Semigroup property, 101
Sender, 182
Sequence, Cauchy, 106
complex-valued, 106
Sequential decision activity, 228
Service time, 175
Sheldon method, 396
Shift operator, 20
Shifting theorem, Heaviside’s, 78, 88
Shunt obstacle, 293
Simplex method in linear programming, 222
Slow growth, function, 25
finite dimensional, 103–104
Space, Hilbert, 104–114
complete, 106
subspace, 107
vector, 10
of convolution quotients, 13
of infinitely differential functions, 24, 25
linear, 57
Spectral norm, 387
Spectral representation of operator, 62
Spectrum, 328
Stable method, 399
Stagnation enthalpy, 368
Standard form of linear programming problem, 222
Star product, 286
dissipative, 292
State vector, 194
Stationary-phase method, 151, 152
Statistical equilibrium (see Steady state)
Statistical sampling, 275
Steady state, 167, 177, 178, 180
Steepest-descent method, 152–159
Steiner symmetrization, 424, 437, 439, 440
Stieltjes integral, 8
Stirling approximation, 137
Stochastic control process, 202–204
Stream function, 365
Streamline, 365
Stripper in oil refinery, 216
Struve function, 92
Sturm-Liouville problem, 95, 341
of Hilbert space, 107
closed, 107
maximal negative, 118, 123, 124
maximal positive, 118
Successive-line overrelaxation, 375
Successive-overrelaxation method, 375, 388–390, 403, 410
Supercritical system, 274
Superposition theorem, Duhamel’s, 79
Support of distribution, 30
Symmetrization, 424–427, 439, 440
alternative, 435
of hill, 430
Symmetrization, line, 424, 425, 437, 440
repeated, 429
in space, 427
Taxicab-stand example, 181
Terminal control, 196
action on, 25
Theory, of elasticity, 431
of games, 202–204
pursuit, 204
Thermal-ignition problem, 412, 413
Thermal oil cracker, 216
Time average, 278–280
Titchmarsh’s theorem, 11
To-and-fro method, 396
Topology, graph, 108
Torsion problem, 407
Traffic, incoming, 175
Trajectory, 202
Transcendental function, 421
Bessel, 70
of derivatives, 78, 88, 89, 96
complex, 69
exponential, 69
multiple, 75
Fourier cosine, 70, 73, 74, 77
Fourier sine, 70, 73, 74, 77, 97
Fourier-Bessel, 70
annular, 96
integral, 68
Laplace, 15, 27, 66, 69, 77–89
of derivatives, 78
inverse of, 54–56, 74, 75, 80–82, 112
Legendre, 96
composite, 272
Euler, 145
generating, 271
iterate of, 273
quadratic, 278
Steiner symmetrization, 424
Transmission coefficient, 283, 319
Transmission line, 282
lossless, 287
passive, 287
Transpose of matrix, 113
Triangle inequality, 105
Truncation error, 357
Two-line overrelaxation method, 375
Two-sided invariant, 328
Unit element of ring, 10
Unit impulse, 5
Value of distribution on interval, 29
Van der Pol equation, 206
Variational problem, 196–209
Vector, 104
complex row, 322
length, 322
inner product, 105
norm, 105
orthogonal, 107
Poynting, 117
state, 194
sum, 104
zero, 105
Vector product with scalar, 104
Vector projection, 107
Vector space, 10, 104–110, 330
of convolution quotients, 13
of infinitely differentiable functions, 24, 25
Vector space, of infinitely differentiable functions, convergence in, 25
linear, 57
Vector-valued distribution, 48
Velocity potential, 365
variance, 181
Waiting time, 174–176
Watson’s lemma, 82, 86, 141, 148
Fourier transform, 89
generalized solution, 8
Laplace transform, 82
semigroup methods for, 102, 104, 115
weak solution, 8
Well-formulated problem, 100, 101, 348, 360
Weyl’s theorem, 278
WKBJ (Wentzel, Kramers, Brillouin, Jeffreys) method, 133
Zero matrix, 113
Zero vector, 105