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2 No lecture on Wed February 8th Thursday 9 th Feb 14: :00 Thursday 9 th Feb 14: :00
3 The Maximum Principle: A Reminder
4 Example k F(k) capital consumption production function depreciation rate
5 Example Solve for c Two differential equations in k,π
6 Example Another way: differentiate
7 Example Another way: X X
8 Example Another way: k c No t !!!!!!
9 Example Another way: k c k’
10 Example Another way: k c k’ k*
11 Example Another way: k c k*
12 Example Another way: k c k*
13 Example Another way: k c k*
14 Example Another way: k c k*
15 Example Another way: k c k* Stationary point
16 Example Another way: k c k*
17 Example Another way: k c k*
18 Example k c k* k(0),c(0) ???? k(0), is given k(0) c(0), is chosen c → 0 k → 0
19 Example k c k* k(0),c(0) ???? k(0), is given k(0) c(0), is chosen c → 0 k → 0
20 Richard E. Bellman Another approach to dynamic programming
21 Another approach to dynamic programming For a given time τ < T define the problem:
22 Another approach to dynamic programming Lagrange: (equating the derivative w.r.t. z t to 0 ) But: ????? ?????? ?
23 Another approach to dynamic programming But: ????? ?????? ? The Lagrangian of the original problem:
24 Another approach to dynamic programming But: ????? ?????? ? The Lagrangian of the original problem:
25 Another approach to dynamic programming but this is the (first order) condition for maximizing the Hamiltonian
26 Another approach to dynamic programming Calculating the Bellman value functions is equivalent to the maximum principle (Hamiltonian)
27 Another approach to dynamic programming
28 Another approach to dynamic programming Backwards Induction
TWO STEP EQUATIONS 1. SOLVE FOR X 3. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE 2. DO THE ADDITION STEP FIRST.
25 seconds left….. 24 seconds left….. 23 seconds left…..
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Mathe III Lecture 2 Mathe III Lecture 2. 2 _____________________________________ Montag 8.00 – 9.30 Uhr – HS H Montag Uhr – HS H Mittwoch.
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MULTIPLICATION EQUATIONS 1. SOLVE FOR X 3. WHAT EVER YOU DO TO ONE SIDE YOU HAVE TO DO TO THE OTHER 2. DIVIDE BY THE NUMBER IN FRONT OF THE VARIABLE.
© 2012 National Heart Foundation of Australia. Slide 2.
Solve Multi-step Equations Students will solve equations with multiple steps (more than two) using distributive property, combining like terms, and inverse.
DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.
Mathematics for Economics Beatrice Venturi 1 Economics Faculty CONTINUOUS TIME: LINEAR DIFFERENTIAL EQUATIONS Economic Applications LESSON 2 prof. Beatrice.
Addition Facts = = =
1 Breadth First Search s s Undiscovered Discovered Finished Queue: s Top of queue 2 1 Shortest path from s.
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