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### p195_exercices_5-6

Course: STAT 601, Spring 2010
School: Texas A&M
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Texas A&M - STAT - 601
Texas A&M - STAT - 601
UCSD - MAE - 101b
NAME:MAE 101B Advanced Fluid Mechanics - Spring 2010 Midterm # 2 - May 3, 2010 50 minutes, open book, open notes, calculator allowed, no cell phones. Write your answers directly on the exam. 30 points total + 5 extra credit points (optional). The exam is
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Copyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California and its InstructorsCopyright 2010 University of Southern California
Native American Education Services - EE - 15.082
15.082 and 6.855JSpring 2003Network OptimizationJ.B. Orlin1WELCOME!x x x x xWelcome to 15.082/6.855J Introduction to Network Optimization Instructor: James B. Orlin (jorlin@mit.edu) TA: Agustin Bompadre (abompadr@mit.edu) Website: sloanspace.mit.ed
Native American Education Services - EE - 15.082
15.082 and 6.855J February 6, 2003Data Structures1Overview of this Lecturexx xA very fast overview of some data structures that we will be using this semester lists, sets, stacks, queues, networks, trees a variation on the well known heap data struc
Native American Education Services - EE - 15.082
15.082 and 6.855J February 11, 2003Lecture 3. Graph SearchBreadth First Search Depth First Search Topological Sort1OverviewToday: Different ways of searching a graph a generic approach breadth first search depth first search data structures to suppor
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15.082 and 6.855J February 13, 2003 Flow Decomposition and Transformations1Flow Decomposition and Transformationsx x x xFlow Decomposition Removing Lower Bounds Removing Upper Bounds Node splitting Arc flows: an arc flow x is a vector x satisfying: Le
Native American Education Services - EE - 15.082
15.082 and 6.855J February 20, 2003Dijkstra's Algorithm for the Shortest Path Problem1Wide Range of Shortest Path ProblemsxSources and Destinations We will consider single source problems in this lecture Properties of the costs. We will consider non-
Native American Education Services - EE - 15.082
15.082 and 6.855JShortest Paths 2: Application to lot-sizing R-Heaps1Application 19.19. Dynamic Lot Sizing (1)xK periods of demand for a product. The demand is dj in period j. Assume that dj &gt; 0 for j = 1 to K. Cost of producing pj units in period j:
Native American Education Services - EE - 15.082
15.082J / 6.855J February 27, 2003The Label Correcting AlgorithmOverview of the Lecturexx xA generic algorithm for solving shortest path problems negative costs permitted but no negative cost cycle (at least for now) The use of reduced costs All pair
Native American Education Services - EE - 15.082
15.082J and 6.855J March 4, 2003Introduction to Maximum Flows1The Max Flow ProblemGij=(N,A)x =ijflow on arc (i,j) capacity of flow in arc (i,j) v source node sink j xij node k xki = 0 for each i s,t= v 0 xij uij for all (i,j) A.u = Maximize s
Native American Education Services - EE - 15.082
15.082 and 6.855J March 6, 2003Maximum Flows 21Network Reliabilityx xCommunication Network What is the maximum number of arc disjoint paths from s to t? How can we determine this number?Theorem. Let G = (N,A) be a directed graph. Then the maximum nu
Native American Education Services - EE - 15.082
15.082 and 6.855J March 11, 2003Max Flows 3 Preflow-Push Algorithms1Review of Augmenting PathsAt each iteration: maintain a flow x Let G(x) be the residual network At each iteration, find a path from s to t in G(x). In the shortest augmenting path alg
Native American Education Services - EE - 15.082
15.082 and 6.855J March 13, 2003 Max Flows 41Overview of today's lecturexVery quick review of Preflow Push Algorithm The Excess Scaling Algorithm O(n2 log U) non-saturating pushes O(nm + n2 log U) running time. A proof that Highest Preflow Push uses O
Native American Education Services - EE - 15.082
15.082 and 6.855J April 1, 2003The Global Min Cut problem1Global Min cutINPUT: A network G = (N, A) OUTPUT: A cut (S, N\S) such that cap(S, N\S) is minimum. Note: We do not assume that there is a source node s and destination node t. Typically, but no
Native American Education Services - EE - 15.082
15.082 and 6.855JApril 3, 2003Introduction to Minimum Cost Flows1The Minimum Cost Flow Problemuij = capacity of arc (i,j). cij = unit cost of shipping flow from node i to node j on (i,j). xij = amount shipped on arc (i,j) Minimize j xij (i,j)A cijxij
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Successive Shortest Path Algorithm and the Capacity Scaling Algorithm for the Minimum Cost Flow Problem1Pseudo-FlowsA pseudo-flow is a &quot;flow&quot; vector x such that 0 x u. Let e(i) denote the excess (deficit) at node i. The infeasibli
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Network Simplex Algorithm1Calculating A Spanning Tree Flow1 1 -6 2 1 3 5 3 6 -4 73A tree with supplies and demands. (Assume that all other arcs have a flow of 0) What is the flow in arc (4,3)?24See the animation.2What would
Native American Education Services - EE - 15.082
15.082 and 6.855The Minimum Cost Spanning Tree Problem1Communications SystemsConsider a communications company, such as AT&amp;T or GTE that needs to build a communication network that connects n different users. The cost of making a link joining i and j
Native American Education Services - EE - 15.082
15.082 and 6.855JReview of Linear Programming1OverviewDescribe LP and IP min cost flow as an LP Graphical solution Basic feasible solutions. Simplex Method Basic feasible solutions in matrix form Duality Note: this will cover lots of material. We will
Native American Education Services - EE - 15.082
15.082J and 6.855JGeneralized Flows1Overview of Generalized FlowsSuppose one unit of flow is sent in (i,j). We relax the assumption that one unit arrives at node j. If 1 unit is sent from i, ij units arrive at j. ij is called the multiplier of (i,j) i
Native American Education Services - EE - 15.082
15.082 and 6.855JLagrangian Relaxation I never missed the opportunity to remove obstacles in the way of unity.-Mohandas Gandhi1On bounding in optimizationIn solving network flow problems, we not only solve the problem, but we provide a guarantee that
Native American Education Services - EE - 15.082
15.082J and 6.855JLagrangian Relaxation 2 xAlgorithms xApplication to LPs1The Constrained Shortest Path Problem2(1,10) (1,2)(1,1) (2,3) (5,7)4(1,7)1(10,1) (2,2)6(10,3)3(12,3)5Find the shortest path from node 1 to node 6 with a transit time
Native American Education Services - EE - 15.082
15.082 and 6.855The Multicommodity Flow Problem1On the Multicommodity Flow Problem O-D versionK origin-destination pairs of nodes (s1, t1), (s2, t2), ., (sK, tK) Network G = (N, A) dk = amount of flow that must be sent from sk to tk. uij = capacity on
Native American Education Services - EE - 15.082
15.082 and 6.855Multicommodity Flows 21On the Multicommodity Flow Problem O-D versionK origin-destination pairs of nodes (s1, t1), (s2, t2), ., (sK, tK) Network G = (N, A) dk = amount of flow that must be sent from sk to tk. uij = capacity on (i,j) sh
Native American Education Services - EE - 15.082
Native American Education Services - EE - 15.082
15.082 and 6.855J Breadth First SearchBreadth first search animation1Initialize2 1 1 pred(1) all Unmark= 0 nodes in N; next := 1 order(next) = Mark node s 1 LIST:= cfw_1 LIST 1 nextBreadth first search animation48 75 3 6912Select a node i in L
Native American Education Services - EE - 15.082
15.082 and 6.855JCycle Canceling Algorithm1A minimum cost flow problem0 2 30, \$7 25 1 25, \$5 20, \$6 3 0 20, \$2 20, \$1 5 -25 10, \$4 0 425, \$22The Original Capacities and Feasible Flow0 2 30,25 25 1 20,0 The feasible flow can be found by solving a m
Native American Education Services - EE - 15.082
15.082 and 6.855JDepth First Search1Initialize2 1 1 pred(1) all Unmark= 0 nodes in N; next := 1 order(next) = Mark node s 1 LIST:= cfw_1 LIST 1 next 1248 75 3 69Select a node i in LIST2 1 1 In depth first search, i is the last node in LIST 3 5
Native American Education Services - EE - 15.082
15.082 and 6.855JDijkstra's Algorithm with simple buckets (also known as Dial's algorithm)1An ExampleInitialize distance labels Initialize buckets. 01 4 2 1 2 3 2 3 3 5244 2 6 Select the node with the minimum temporary distance label. 01 4 5 6
Native American Education Services - EE - 15.082
15.082 and 6.855JFlow Decomposition1begin Initialize while y do begin Select(s, y) Search(s, y) if a cycle C is found then do begin let = Capacity(C, y) Add Flow( , C) to cycle flows Subtract Flow( , C) from y. end if a path P is found then do begin le
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem1Ford-Fulkerson Max Flow23 2 3 1 4 1 151 2 2s4 3tThis is the original network, plus reversals of the arcs.2Ford-Fulkerson Max Flow23 2 3 1 4 1 151 2
Native American Education Services - EE - 15.082
15.082 and 6.855JLabel Correcting Algorithm1An Example 23 1 6 3 4 3 2 43 -2 3Initialize d(1) := 0; d(j) := for j 1015 3 -42 76 In next slides: the number inside the node will be d(j). Violating arcs will be in thick lines.2An Example2
Native American Education Services - EE - 15.082
15.082 and 6.855JMin Global Cut Animation1Initialize11 5 3 3 2 Saturate the arcs out of node 1. Update the residual network4 41 6 1 562Initialize1 5 4 3 2 1 0 4 6 3 5 2 We will never push from node 1 5 again or into node 1. 5 3 3 3 2 Compute di
Native American Education Services - EE - 15.082
15.082 and 6.855JModified Label Correcting Algorithm1The Modified Label Correcting Algorithm 23 1 6 3 4 3 2 53 -2 3Initialize d(1) := 0; d(j) := for j 1 LIST := cfw_1014 3 -42 76 In next slides: the number inside the node will be d(j).2A
Native American Education Services - EE - 15.082
15.082 and 6.855J Network Simplex Animations1Calculating A Spanning Tree Flow1 1 -6 2 1 3 5 3 6 -4 73A tree with supplies and demands. (Assume that all other arcs have a flow of 0) What is the flow in arc (4,3)?242Calculating A Spanning Tree Flow
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Goldberg-Tarjan Preflow Push Algorithm for the Maximum Flow ProblemPreflow Push23 2 3 1 4 1 151 4 2s4 3tThis is the original network, plus reversals of the arcs.Preflow Push23 2 3 1 4 1 151 4 2s4 3tThis is the origi
Native American Education Services - EE - 15.082
15.082 and 6.855J February 25, 2003Radix Heap Animation1An Example from AMO (with a small change)Initialize distance labels Initialize buckets and their ranges. Insert nodes into buckets.2 0 1 3 4 71325 15420 10 20 9 868 15 16 31 32 63352
Native American Education Services - EE - 15.082
15.082 and 6.855JThe Shortest Augmenting Path Algorithm for the Maximum Flow Problem1Shortest Augmenting Path23 2 3 1 4 1 151 4 2s4 3tThis is the original network, plus reversals of the arcs.2Shortest Augmenting Path23 2 3 1 4 1 151 4 2s
Native American Education Services - EE - 15.082
15.082 and 6.855JSpanning Tree Algorithms1The Greedy Algorithm in Action352 2104 48 156 61 1402520 15305 521 11173 37 72The Greedy Algorithm in Action35 352 2 25 2510 104 4 30 30 5 5815 156 6 17 171 140 4020 203 31121 21
Native American Education Services - EE - 15.082
15.082J and 6.855JSuccessive Shortest Path Algorithm1The Original Costs and Node Potentials0 2 7 5 0 1 6 3 040 42125 02The Original Capacities and Supplies/Demands5 2 30 23 1 20 3 -7 25 20 20 10 -2 4255 -193Select a supply node and find t
Native American Education Services - EE - 15.082
15.082 and 6.855J Topological Ordering1Preliminary to Topological SortingLEMMA. If each node has at least one arc going out, then the first inadmissible arc of a depth first search determines a directed cycle.1 4 6 7 3COROLLARY 1. If G has no directe
University of Aarhus - BUS - 1332
Tecumseh (March 1768 October 5, 1813) also Tecumtha or Tekamthi, was a Native American leader ofthe Shawnee and a large tribal confederacy that opposed the United States during Tecumseh's War and the War of 1812. He grew up in the Ohio country during the
University of Aarhus - BUS - 1332
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University of Aarhus - BUS - 1332
First YearFall Semester CHEM 1117. Chemistry for Engineers Laboratory CHEM 1372. Chemistry for Engineers ECE 1100. Introduction to Electrical and Computer Engineering ENGL 1303. Freshman Composition I HIST 1377. The United States to 1877 1 MATH 1431. Cal
University of Aarhus - BUS - 1332
Midterm Study Guide- 1377 Professor Hopkins MW 11:00-12:30, AH 540 See blackboard emails regarding your specific TA. If you need to contact any of us, please do so through blackboard- not personal email. READ THIS FIRSTThis midterm exam is a writing exam
University of Aarhus - BUS - 1332
Final Exam Study Guide- 1377 FA 09 MW Professor Hopkins READ THIS FIRSTThe Final Exam is Monday, December 14, from 5-8 pm, in the lecture auditorium. You must show your UH Student ID when turning in your finished exam, and sign the class roster. Like the
University of Aarhus - BUS - 1332
University of Aarhus - BUS - 1332
Sun 5 am:30 :15 :30 :45MonTuesWedThurFri[42]Sat6 am7 am:15 :30 :458 am:15 :30 :459 am:15 :30 :4510 am:15 :30 :4511 am:15 :30 :4512 PM:15 :30 :451 PM:15 :30 :452 PM:15 :30 :453 PM:15 :30 :454 PM4:15 :30 :455 PM:15 :30 :456
York UK - MGT - mgt600
York UK - MGT - mgt600
York UK - MGT - mgt600
York UK - MGT - mgt600
York UK - MGT - mgt600
York UK - MGT - mgt600
Maryland - MATH - 464
Homework #1 Due: Tuesday, February 2, 20101. (2pt) Compute the following integrals:1 log( x)dx0where log(x) denotes the natural logarithm of x, and 1 x a dx 1 for a fixed parameter a&gt;1. What happens for a=1 ?2. (5pt) Consider the following function:
Maryland - MATH - 464
Homework #2 Due: Tuesday, February 9, 2010Compute the following integrals. Note: Use of Matlab (or any other software) is not permitted. 1 1. x 2 + 1 dx 02.- x21 dx , for a fixed real parameter a&gt;0 + a2 1 dx , for a fixed real parameter a&gt;0 + 1)( x
Maryland - MATH - 464
Homework #3 Due: Thursday, February 25, 2010Compute the Fourier coefficients, and expand in Fourier series the following 1-periodic functions. Note: Use of Matlab (or any other software) is not permitted. 1.2(1 - 2 x ), for 0 &lt; x &lt; 12.26(1 - 6 x +
Maryland - MATH - 464
Homework #4 Due: Thursday, March 4, 2010Note: Use of Matlab (or any other software) is not permitted. I. Compute the Fourier transform of the following functions (1-8): 1. 1 , 1 x 2 f ( x) = 0 , otherwise 0 , x &lt; -3 ( x - 3) / 2 , - 3 x &lt; -1 f ( x) = 1 ,
Maryland - MATH - 464
Homework #5 Due: Thursday, March 11, 2010Note: Use of Matlab (or any other software) is not permitted. I. (Exercise 3.23) Let f be a suitably regular function on R with the Fourier transform F. What can you deduce about F if you know that:1.- f ( x)d