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Inner Product on Matrices), 41 a (Closure of A), 308 a ≼ B (Matrix Ordering Index A : B (inner product on matrices), 41 K∞ (asymptotic cone), 19 A • B (inner product on matrices), 41 K ∗ (dual cone), 19 A (closure of A), 308 L(X,Y ) (linear maps X → Y ), 312 A . B (matrix ordering), 184 L∞(), 311 A∗(adjoint operator), 313 L p(), 311 BX (open unit ball), 17 λ (Lebesgue measure), 317 C(; X) (continuous maps → X), λmin(B) (minimum eigenvalue of B), 315 142, 184 ∞ Rd →∞ C0 ( ) (space of test functions), 316 liminfn xn (limit inferior), 309 co(A) (convex hull), 314 limsupn→∞ xn (limit superior), 309 cone(A) (cone generated by A), 337 M(A) (space of measures), 318 d D(R ) (space of distribution), 316 NK (x) (normal cone), 19 Dα (derivative with multi-index), 322 O(g(s)) (asymptotic “big Oh”), 204 dH (A, B) (Hausdorff metric), 21 ∂ f (subdifferential of f ), 340 δ (Dirac-δ function), 3, 80, 317 P(X) (power set; set of subsets of X), 20 + δH (A, B) (one-sided Hausdorff (U) (strong inverse), 22 − semimetric), 21 (U) (weak inverse), 22 diam F (diameter of set F), 319 K (projection map), 19, 329 n divσ (divergence of σ), 234 R+ (nonnegative orthant), 19 dµ/dν (Radon–Nikodym derivative), S(Rd ) (space of tempered test 320 functions), 317 domφ (domain of convex function), 54, σ (stress tensor), 233 327 σK (support function), 328 epi f (epigraph), 19, 327 sup(A) (supremum of A ⊆ R), 309 ε (strain tensor), 232 TK (x) (tangent cone), 19 −1 f (E) (inverse set), 308 (u, v)H (inner product), 17 f g (inf-convolution), 348 u, v (duality pairing), 17 graph (graph of ), 21 W m,p() (Sobolev space), 322 Hess f (Hessian matrix), 81 X\A (set difference), 308 H m() (Sobolev space), 322 X (dual space), 17, 312 IK (indicator function), 63, 327, 328 (natural map), 18, 313 inf(A)(infimum of A ⊆ R), 309 xn x (weak convergence), 314 ∗ int A (interior of A), 308 xn x (weak* convergence), 314 JX (duality map), 18, 313 χE (characteristic function), 318 K ◦ (polar cone), 19 x ◦ y (Jordan algebra), 41 381 382 Index x ◦ y (Hadamard or componentwise complete metric space, 308, 309 product), 41 cone, 19, 328 x ⊗ y (tensor product), 300 asymptotic, 19 convex, 19 absolute continuity, 105, 107, 125, 320 dual, 19, 331 absorbing set, 312, 349 ice cream, 40 action, 209 Lorentz, 40 active set, 139, 162 normal, 19, 122, 337 adjoint operator, 88, 313 of semidefinite matrices, 41 Alaoglu’s theorem, 25–28, 71, 104, 114, pointed, 19, 127, 330 134, 171, 227, 246, 254, 257, polar, 19, 331 267, 279, 281, 314, 318, 339 polygonal, 86 algebraically stable, 296 polyhedral, 19, 38, 66 Amontons, Guillame, 6 recession, 19, 26, 46, 126, 127, 337 Arzela–Ascoli theorem, 107, 154, 159, self-dual, 19, 40, 331, 333 227, 315 strongly pointed, 19, 128, 331 Asplund space, 351 symmetric, 41 asymptotic cone, 19 tangent, 19, 335 constitutive relation, 233 B-stable, 296 constraint qualification, 31, 123, 344 Baire category theorem, 309, 341, 349 linear independence (LICQ), 31 Bellman, Richard, 354 bipolar junction transistor (BJT), 193 Mangasarian–Fromowitz (MFCQ), Bochner integral, 319 31 Bohl distribution, 86, 162 Slater, 31, 68, 345 Borel measurable, 318 continuation method, 37 Borel set, 29, 318 contraction mapping theorem, 52, 117, Boston traffic equilibrium, 199 142, 325, 354 Bouligand generalized gradient, 351 convergence bounded variation, 125, 319 strong, 314 Butcher tableau, 294 weak, 314 convex, 327 Carathéodory, Constantin, 353 cone, 19 Carathéodory function, 30 function, 18 catching-up algorithm, 125, 131 function, proper, 19 Cauchy sequence, 309 hull, 102 chattering, 1 projection, 19 Clarke regular, 351 series closed, 348 closed graph, 21 set, 18 closed set, 307 convolution, 358 coefficient of restitution, 3, 85, 143, 211 convolution complementarity problem coercivity, 49, 339 (CCP), 13, 141–144, 167–178, C(), 311 185, 239, 240, 271, 273 compact operator, 53, 89, 312 copositive, 35, 221, 224 compact set, 308 K -copositive, 39 compensated compactness, 257 plus, 35 complementarity problem (CP), 3, strictly, 35, 292 30–42 strongly, 40, 46, 47, 152, 168, 172 Index 383 core, 312 elastic rod, 5 Coriolis forces, 209 elasticity, 233 Cottle, Richard, 31 elliptic operator, 42, 91, 243, 282 Coulomb, Charles A., 6 energy Coulomb friction, 6, 211 kinetic, 208 Cournot equilibrium, 196 potential, 208 covering vector, 34 energy-based impact law, 215 epigraph, 54, 327 d’Alembert solution, 240 equicontinuity, 107, 315 da Vinci, Leonardo, 6 equi-integrable, 103, 107 Dantzig, George, 31 equivalent norms, 310 degree theory, 49 Erdmann’s condition, 230 Delassus, Étienne, 217 essentially bounded function, 311 dense, 308 Euclidean Jordan algebra, 41 dense operator, 88 Euler, Leonhard, 6 differential complementarity problem Euler–Bernoulli beam, 250 (DCP), 14, 79, 98, 124, 143, Euler–Lagrange equations, 209 152, 161, 210 Euler’s method, 106 differential games, 196 evolution triple, 88 differential inclusion, 8, 83, 101, 217 differential measure, 125, 319 Fenchel dual, 54, 145, 161, 212, 342, differential variational inequality (DVI), 348 77–92, 146–205, 213 Fichera, Gaetano, 42 index-one, 83 Filippov’s lemma, 103, 107 index-two, 84 Fitzpatrick function, 57 index-zero, 82 fixed point, 325 mixed-index, 151 flip-flop, 194 pure index-one, 151 Fourier transform, 89, 90, 275, 322, 358 diode, 9, 12, 178–192 fractional derivative, 142 Dirac-δ function, 3, 80, 84, 95, 117, 126, Fréchet differentiable, 350 142, 317 friction, 5, 211 Dirichlet to Neumann operator, 259 anisotropic, 211 distribution, 80, 86, 316 coefficient, 6, 7 tempered, 359 cone, 213 div-curl lemma, 257 Coulomb, 6, 81, 101, 109, 110, domain, 327 130, 151, 211, 228, 271, 285 drift, 289 elastic body, 235 dual jamming, 226 Fenchel, 54, 145, 161, 212, 342, nonlocal, 239, 272 348 Painlevé’s paradox, 217 dual cone, 19, 31, 39, 192, 250, 251, torque, 212 275, 331 Tresca, 238 dual space, 17, 312 two-coefficient model, 6 duality variational inequality (VI), 212 weak, 343 friction coefficients, 6 duality gap, 344 Fritz John condition, 347 Dunford–Pettis theorem, 103, 107 Frobenius inner product, 41 384 Index Fσ set, 309 reduction, 283, 285 function three, 82, 86 convex, 54 two, 81, 84, 143, 207 functional, 312 zero, 82, 142, 147, 287, 289, 298 indicator function, 63, 119, 212, 288, Galerkin method, 252, 257 327, 328 % function, 274 inf-convolution, 58, 348 gap function, 235 infimum, 309 Gateaux differentiable, 350 inner product, 311 Gδ set, 309 integrable Gelfand triple, 88, 111, 117 function, 77, 80, 83, 102, 126, 142, generalized complementarity problems 320 (GCPs), 31 selection, 103 generalized gradient, 350 set-valued function, 103 generalized Jacobian, 351 interpolation space, 253 graph, 14, 179 inverse image directed, 197 strong, 22 Gronwall’s lemma, 109, 113, 354 weak, 22 GUS(K ), 41 John, Fritz, 347 Hadamard product, 41 Jordan algebra, 166 Hahn–Banach theorem, 327, 341 JX , duality map, 18, 313 Hausdorff metric, 21, 124 heat equation, 87, 118 Kakutani fixed point theorem, 326 Heaviside function, 240, 281 Karush–Kuhn–Tucker (KKT) condition, Heaviside model, 12 4, 8, 31, 68, 210, 344, 347 Heaviside, Oliver, 12 kinetic energy, 208 hemicontinuous, 21 Kotel’nikov, S., 6 H m(), 322 Kronecker δ, 234 Hölder continuity, 323 Kuhn–Tucker condition, 4, 8, 344 homotopy, 37, 50 Ky Fan theorem, 326 hyperelastic, 238, 271 hypergraph, 179 2, 141 Lagrange multipliers, 4, 210, 347 ice cream cone, 40 Lagrangian, 4 ice skating, 212 Lamé parameters, 234 impact law Laplace transform, 137, 358 energy-based, 215 Laurent series, 138 Newton’s, 213 Lebesgue decomposition, 126 Poisson’s, 214 Lebesgue measure, 102 index, 8, 10, 13, 285 Lemke, Carlton, 31 convolution complementarity Lemke’s method, 31, 34, 37, 221 problem (CCP), 141 lemma fractional, 144 div-curl, 257 linear complementarity system Filippov’s, 103, 107 (LCS), 139 Gronwall’s, 109, 113, 354 one, 81, 83, 124, 143, 145, 147, Mazur’s, 26, 114, 120, 314 288, 291, 298 reversibility, 33 Index 385 Young’s, 173, 321 time stepping, 293 Zorn’s, 55 metric, 307 Leray–Schauder theorem, 325 metric space, 307 lexicographically positive, 137 minimal spanning tree (MST), 180 linear complementarity problem (LCP), Minty’s theorem, 57 30–38, 40, 139, 161, 219–221, mixed complementarity problem, 8 223, 291 monotone linear complementarity system (LCS), strict, 50 10, 81, 86, 189 strong, 151 linear programming, 31 monotone operator, 55 Lipschitz boundary, 323 Moore–Penrose pseudoinverse, 39 Lipschitz continuity, 308 Morse–Sard theorem, 38 Lorentz cone, 40 multi-index, 322, 359 lower semicontinuity, 19, 21, 43, 54, 327 L p(), 311 natural map, 18, 45, 332 LS(K ), 41 neighborhood, 308 Lucas critique, 196 network, 14, 179 Neumann to Dirichlet operator, 240, mass matrix, 4, 208 258, 259 maximal monotone operator, 84, 91, 340 Newton’s impact law, 213 maximum dissipation principle, 8, 211 Newton’s law of restitution, 85 Mazur’s lemma, 53, 114, 120, 314 Newton’s laws, 237 measurable norm, 310 equivalent, 310 strongly, 29 normal cone, 19, 62, 64, 122, 337 weakly, 29 normal map, 44 measurable function, 103 measurable selection, 30, 102, 103, 106, obstacle, 5 107 thick, 242 measure, 125, 316, 317 thin, 239 bounded variation, 319 obstacle problem, 42, 118, 119 differential, 125 one-sided Lipschitz continuity, 108 Lebesgue, 317 open set, 307 variation, 126 measure differential inclusion (MDI), P-function, 149 125, 208, 217 P-matrix, 36, 139, 143 strong definition, 127 Painlevé, Paul, 217 weak definition, 127 Painlevé’s paradox, 217, 231 meromorphic function, 139 parabolic variational inequality (PVI), method 16, 144–146, 161, 238 continuation, 37 particle, 207 Euler, 294 passive system, 190 homotopy, 37 Picard iteration, 116, 142, 353
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