Principles of Mathematics in Operations Research
8.5 Farkas' Lemma 113 &yTB cTB^yT cTBB~x & yTb = cTBB~xb = cTx\ Furthermore, this choice of y is optimal, and the strong ...
Proof. Ax > b, x > Either \A-I\ X AT -I y > 0,yTb < 0. 114 8 Linear Programming » -» Ax - Iz = b,z>9. has a nonne ...
8.5 Problems 115 Either P4 is feasible, or D4 is unbounded. For D4 to be unbounded, we must have bTy < 0. Thus, either Ax > ...
116 8 Linear Programming combination of extreme points plus the canonical combination of extreme rays (if any) of P. e) Let the ...
Web material Fig. 8.3. Starting bfs solution for our multi-commodity flow instance b) Let Vk be the set of paths from source nod ...
118 8 Linear Programming http://en.wikipedia.org/wiki/Farkas's_lemma http: //en. wikipedia. org/wiki/Linear_programming http://m ...
8.6 Web material 119 http://www.math.Chalmers.se/Math/Grundutb/CTH/tma947/0506/ lecture9.pdf http://www.math.kth.se/optsyst/rese ...
9 Number Systems In this chapter, we will review the basic concepts in real analysis: order re- lations, ordered sets and fields ...
122 9 Number Systems bounded if E is both bounded above and below. Example 9.1.5 A = {p G Q : p >- 0,p^2 •< 2} is bound ...
9.2 Fields Claim (ii): a is the greatest of the lower bounds. Proof (ii): Show if a -< /?, /3 £ 5 => /? is not a lower bou ...
124 9 Number Systems Proposition 9.2.4 In a field F, the following properties hold: (a) x + y = x + z=>y = z (cancelation law ...
9.3 The Real Field Remark 9.2.8 C with usual + and • is a field. But it is not an ordered field. If x — i then i^2 = — 1 y 0, he ...
126 9 Number Systems Theorem 9.3.4 Va; G R, x y 0, Vn G N 3 a Mrogtte y G R, y X 0 9 yn = x. Proof. [Existence]: Given x X 0, n ...
9.4 The Complex Field 127 Proof. Let a = a^1 /", /? = b^1 '^71 => an = a, (3n = b => (a/?)" = an(3n = ab y 0 and n"^1 root ...
128 9 Number Systems Proposition 9.4.1 Let z,w G C. Then, (a) z ^ 0 => \z\ > 0 and |0| = 0. W 1*1 = N- (c,) |^u)| = |z||i« ...
9.6 Countable and Uncountable Sets 129 Definition 9.5.2 An equivalence relation in X is a binary relation (where ~ means equival ...
130 9 Number Systems Define f:N^Q+, /(l) = 1, /(2) = |, /(3) = 2, /(4) = |, Cantor's Counting Schema Another Counting Scheme Fig ...
9.6 Countable and Uncountable Sets 131 Proposition 9.6.6 If s = {xi,i £ 1} is a countable class of countable sets, then Ui£iXi i ...
132 9 Number Systems 1 !• ^ %- -%f y >y y ^ Fig. 9.2. Uncountability equivalence of (a,b) and (0,1) -1 +1 Fig. 9.3. The corre ...
9.6 Problems 133 Definition 9.6.13 Roughly speaking, the cardinality of a set (or cardinal number of a set) is the number of ele ...
«
2
3
4
5
6
7
8
9
10
11
»
Free download pdf