Principles of Mathematics in Operations Research

278 Solutions li norm: Li-tJi-tKi-tGi->Ii->Hi->Fi-+Ai-*Cy-*B*->Di-*E with length 38; l 2 norm: fHC^/nJ^i/^F^i^C^Bi-y ...

Solutions 279 Problems of Chapter 11 11.1 a) (=>): Let e > 0 and XQ be given. Let b — /(xo) — e. Then, by assumption, the ...

280 Solutions Problems of Chapter 12 12.1 Use the Mean Value Theorem: h : R >—» R is nondecreasing if h'(x) > 0. y<x=&g ...

Solutions 281 f(x*+h)^f(x*)+ 1 -h T V 2 f(x*)h. Suppose that V^2 /(x*) is not positive semi-definite. Then, 3«eln 9 vTV^2 f(x*)v ...

282 Solutions -0.5 -0.5 -1 -1 Fig. S.24. Plot of f(xux 2 ) = xi + 2x\ + 2Ax\ + x% + 12x| Then, V/ 0.194444 0.222222 x{2) = V^2 / ...

Solutions 283 V^2 / 0.00000398 0.00000002 48.00005 0 0 24 Then, x w = 0.00000398" 0.00000002 Finally, V/ ( 1 48.00005 1.98 ...

284 Solutions Problems of Chapter 13 13.1 Let there be given two series A = 2~] Uk and B — \ vk with nonnegative terms. (a) // W ...

Solutions 285 Thus, it is convergent. b) £r f^> where a > 0: It is convergent for x = 0. Let us assume that 0 < x < ...

286 Solutions r <n objects out of n objects is (™). Thus, ar = (™). We cannot choose more than n objects; that is ar = 0, r & ...

Solutions 287 (x + y)n = Yf(^)(x 1 + -.. + xl)ixti i=0 . „ ,t/ *-^ \«i,••-,«/ i\-\ Yi\ — i Y^ I • ". • ix^-x^x (^71) »i «i™«'+i ...

288 Solutions Similarly, p(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^6 p(x) = x^6 + x^5 + 1 — x^6 (1 - x^6 )p(x) = 1 + ...

Solutions 289 =>an = 2n+3n. 13.4 a) The left hand side of the following constraint represents the complementary survival prob ...

290 Solutions Problems of Chapter 14 14.1 y"(t) - y(t) = e2t & s^2 r](s) - 2s - r)(s) = <& r?(s)(s^2 - 1) = r + 2s. s ...

Solutions 291 dt^2 = s^2 r](s) - sy(0) - y (0) = s^2 rj(s) - 100s + 25, since %\t=0 = -0.3(50) - 0.1(100) = -25. Moreover, V &am ...

292 Solutions Thus, Finally, Fioo = x(99) = 354 224 848179 261915 075 ...

Index affine combination, 15 hull, 16 aleph, 133 Archimedean Property, 125 axiom, 4, 123, 138 Balzano-Weierstrass, 150 basis, 16 ...

294 Index differential equations, 60, 62-63, 65, 191-196, 201, 202, 228, 290 dodecahedron, 100, 244 eigen value, 54-65, 74-76, 8 ...

Index 295 unboundedness, 106, 251, 252 Markov process, 61, 87 matrix asymmetric, 83 basis, 39 block diagonal, 58 column space, 2 ...

296 Index trigonometric, 184 polytope, 96, 99-100, 115, 241-244, 246 projection, 37-43 proof, 4 proof making, 5-9, 206-209 combi ...

column space, 26, 28, 44, 211-215 direct sum, 59 Euclidean, 128, 147 half space, 96 left null space, 27, 28, 44, 211-215 null sp ...