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Example 5 Eva, Wong, Yumi and Kim are students who are staying in a backpacker's hostel. Because they! languages they can have problems communicating. The situation they have to deal with is that: a) b) • Eva speaks English only . . . Yumi speaks Japanese only Kim speaks Korean only Wong speaks English, Japanese and Korean. Summarise this information in a network diagram. Construct a communication matrix, C, to represent the situation, where rows represent the speakers and columns the receivers. c) Find C¹.

Example 5 Eva, Wong, Yumi and Kim are students who are staying in a backpacker's hostel. Because they! languages they can have problems communicating. The situation they have to deal with is that: a) b) • Eva speaks English only . . . Yumi speaks Japanese only Kim speaks Korean only Wong speaks English, Japanese and Korean. Summarise this information in a network diagram. Construct a communication matrix, C, to represent the situation, where rows represent the speakers and columns the receivers. c) Find C¹.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Example 5 Eva, Wong, Yumi and Kim are students who are staying in a backpacker's hostel. Because they speak different languages they can have problems communicating. The situation they have to deal with is that: a) b) d) e) 8) f) . . . c) Find (². . Eva speaks English only Yumi speaks Japanese only Kim speaks Korean only Wong speaks English, Japanese and Korean. Summarise this information in a network diagram. Construct a communication matrix, C, to represent the situation, where rows represent the speakers and columns the receivers. Interpret the element (1, 1) in C² Can you find an example of a redundant link in the previous matrix? Can you think of a legitimate situation where a redundant link may actually be feasible? Calculate T for our communications matrix. (T=C+C².) 5| Page Example 6 Four fire towers T1, T2, T3 and T4, can communicate with one another as shown in the diagram opposite. In this diagram an arrow indicates that a direct channel of communication exists between a pair of fire towers. The matrix C² is shown opposite. d Explain the meaning of the 1 in row T3, column T1 e How many of the two-step communication links shown in the matrix C² are redundant? TI For example, a person at tower 1 can directly communicate with a person in tower 2 and vice versa. The communication matrix C can be also used to represent this information. a Explain the meaning of a zero in the communication matrix. b Which two towers can communicate directly with T2? c Write down the values of the two missing elements in the matrix. C = C²= 0 0 72 T1 T2 T3 T4 0 1 10 1 0 A 0 11 1 T1 1 0 1 0 73 T2 T3 T4 0 1 0 2 10 2 0 0 TI 0 72 1 T3 0 T4 071 1 T2 0 T3 1 T4 f Construct a matrix that shows the total number of one and two-step communication links between each pair of towers. g Which of the four towers need a three-step link to communicate with each other? T4 6| Page
Transcribed Image Text:Example 5 Eva, Wong, Yumi and Kim are students who are staying in a backpacker's hostel. Because they speak different languages they can have problems communicating. The situation they have to deal with is that: a) b) d) e) 8) f) . . . c) Find (². . Eva speaks English only Yumi speaks Japanese only Kim speaks Korean only Wong speaks English, Japanese and Korean. Summarise this information in a network diagram. Construct a communication matrix, C, to represent the situation, where rows represent the speakers and columns the receivers. Interpret the element (1, 1) in C² Can you find an example of a redundant link in the previous matrix? Can you think of a legitimate situation where a redundant link may actually be feasible? Calculate T for our communications matrix. (T=C+C².) 5| Page Example 6 Four fire towers T1, T2, T3 and T4, can communicate with one another as shown in the diagram opposite. In this diagram an arrow indicates that a direct channel of communication exists between a pair of fire towers. The matrix C² is shown opposite. d Explain the meaning of the 1 in row T3, column T1 e How many of the two-step communication links shown in the matrix C² are redundant? TI For example, a person at tower 1 can directly communicate with a person in tower 2 and vice versa. The communication matrix C can be also used to represent this information. a Explain the meaning of a zero in the communication matrix. b Which two towers can communicate directly with T2? c Write down the values of the two missing elements in the matrix. C = C²= 0 0 72 T1 T2 T3 T4 0 1 10 1 0 A 0 11 1 T1 1 0 1 0 73 T2 T3 T4 0 1 0 2 10 2 0 0 TI 0 72 1 T3 0 T4 071 1 T2 0 T3 1 T4 f Construct a matrix that shows the total number of one and two-step communication links between each pair of towers. g Which of the four towers need a three-step link to communicate with each other? T4 6| Page
Dominance matrices Dominance matrix is a square binary matrix in which 1s represent one-step dominance between members of a group. In many group situations, certain individuals are said to be dominant. Problems of identifying dominant individuals in a group can be analysed using the same approach we used to analyse communication networks. This method is used to rank teams or players who are playing competition against each other. D represents one-step dominance D² represents two-step dominance Total dominance scores, T = D + D² Example 7 Five tennis players - Anna, Birgit, Cas, Di and Emma-compete in a round-robin tournament (each player plays every other player once). Who is the best player? Another way of asking this might be: Who is the DOMINANT player? The results are: Anna defeated Cas and Di Birgit defeated Anna, Cas and Emma Cas defeated Di . Di defeated Birgit . Emma defeated Anna, Cas and Di. a) Summarise this information in a network diagram, with arrows moving from the winner to the loser. . b) Construct a one-step dominance matrix, D, to represent the situation, where players are listed in alphabetical order. Enter a '1' in each ROW to show who the player on the left won against. Find the DOMINANCE SCORE for each player by summing the values in each row. A B D = from C D E to A BCD 0 0 1 1 E 0 This is a one step Domination matrix because it tells us who defeated who, or who dominated over who, in their individual game. If we add the elements in each row, we get the total number of wins (one-step dominances) of each player. 7| Page Now we can also take into account two step dominances in order to rank the players. A two step dominance occurs when for example A defeats C who then defeats D. A is said to have a two step dominance over D. We can find the two-step dominance matrix by squaring D. Once we have done this, we can add together the one and two-step dominances to get a total dominance score. Find D², the two-step dominance matrix and also find the total for each row. c) d) from Total dominance matrix = T =D+D² 8 C A 8 C D E Calculate the total dominance score for each player by first finding T and hence rank the players. Example 8 The graph shows the one-step dominances between four farm dogs, Kip, Lab, Max and Nim. In this graph, an arrow from Lab to Kip indicates that Lab has a one-step dominance over Kip. From this graph, it can be concluded that Kip has a two-step dominance over A Max only. B Nim only. D all of the other three dogs. E none of the other three dogs. Nim C Lab and Nim only. MAX 8 | Page
Transcribed Image Text:Dominance matrices Dominance matrix is a square binary matrix in which 1s represent one-step dominance between members of a group. In many group situations, certain individuals are said to be dominant. Problems of identifying dominant individuals in a group can be analysed using the same approach we used to analyse communication networks. This method is used to rank teams or players who are playing competition against each other. D represents one-step dominance D² represents two-step dominance Total dominance scores, T = D + D² Example 7 Five tennis players - Anna, Birgit, Cas, Di and Emma-compete in a round-robin tournament (each player plays every other player once). Who is the best player? Another way of asking this might be: Who is the DOMINANT player? The results are: Anna defeated Cas and Di Birgit defeated Anna, Cas and Emma Cas defeated Di . Di defeated Birgit . Emma defeated Anna, Cas and Di. a) Summarise this information in a network diagram, with arrows moving from the winner to the loser. . b) Construct a one-step dominance matrix, D, to represent the situation, where players are listed in alphabetical order. Enter a '1' in each ROW to show who the player on the left won against. Find the DOMINANCE SCORE for each player by summing the values in each row. A B D = from C D E to A BCD 0 0 1 1 E 0 This is a one step Domination matrix because it tells us who defeated who, or who dominated over who, in their individual game. If we add the elements in each row, we get the total number of wins (one-step dominances) of each player. 7| Page Now we can also take into account two step dominances in order to rank the players. A two step dominance occurs when for example A defeats C who then defeats D. A is said to have a two step dominance over D. We can find the two-step dominance matrix by squaring D. Once we have done this, we can add together the one and two-step dominances to get a total dominance score. Find D², the two-step dominance matrix and also find the total for each row. c) d) from Total dominance matrix = T =D+D² 8 C A 8 C D E Calculate the total dominance score for each player by first finding T and hence rank the players. Example 8 The graph shows the one-step dominances between four farm dogs, Kip, Lab, Max and Nim. In this graph, an arrow from Lab to Kip indicates that Lab has a one-step dominance over Kip. From this graph, it can be concluded that Kip has a two-step dominance over A Max only. B Nim only. D all of the other three dogs. E none of the other three dogs. Nim C Lab and Nim only. MAX 8 | Page
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