We consider a system of reservoirs connected to each other through pumps. An example system is shown below in Figure 1, represented as a graph. Each node in the graph is marked with a letter and represents a reservoir. Each edge in the graph represents a pump which moves a fraction of the water from one reservoir to the next at every time step. The fraction of water moved is written on top of the edge. قين Figure 1: Pump system We want to prove the following theorem. We will do this step by step. Theorem: Consider a system consisting of k reservoirs such that the entries of each column in the system's state transition matrix sum to one. If s is the total amount of water in the system at timestep n, then total amount of water at timestep n+1 will also be s. (a) Rewrite the theorem statement for a graph with only two reservoirs. (b) Since the problem does not specify the transition matrix, let us consider the transition matrix A a11 a12 and the state vector X[n] = [x]]. Write out what is "known" or what is given to you in [a21 a22] [x₂[n]] the theorem statement in mathematical form. Hint: In general, it is helpful to write as much out mathematically as you can in proofs. It can also be helpful to draw the transition graph. (c) Now write out what is to be proved in mathematical form. (d) Prove the statement for the case of two reservoirs. (e) Now use what you learned to generalize to the case of k reservoirs. Hint: Think about A in terms of its columns, since you have information about the columns.
We consider a system of reservoirs connected to each other through pumps. An example system is shown below in Figure 1, represented as a graph. Each node in the graph is marked with a letter and represents a reservoir. Each edge in the graph represents a pump which moves a fraction of the water from one reservoir to the next at every time step. The fraction of water moved is written on top of the edge. قين Figure 1: Pump system We want to prove the following theorem. We will do this step by step. Theorem: Consider a system consisting of k reservoirs such that the entries of each column in the system's state transition matrix sum to one. If s is the total amount of water in the system at timestep n, then total amount of water at timestep n+1 will also be s. (a) Rewrite the theorem statement for a graph with only two reservoirs. (b) Since the problem does not specify the transition matrix, let us consider the transition matrix A a11 a12 and the state vector X[n] = [x]]. Write out what is "known" or what is given to you in [a21 a22] [x₂[n]] the theorem statement in mathematical form. Hint: In general, it is helpful to write as much out mathematically as you can in proofs. It can also be helpful to draw the transition graph. (c) Now write out what is to be proved in mathematical form. (d) Prove the statement for the case of two reservoirs. (e) Now use what you learned to generalize to the case of k reservoirs. Hint: Think about A in terms of its columns, since you have information about the columns.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![3. Properties of Pump Systems - II
Learning Objectives: This problem builds on the pump examples we have been doing, but is meant to help
you learn to do proofs in a step by step fashion. Can you generalize intuition from a simple example?
We consider a system of reservoirs connected to each other through pumps. An example system is shown
below in Figure 1, represented as a graph. Each node in the graph is marked with a letter and represents a
reservoir. Each edge in the graph represents a pump which moves a fraction of the water from one reservoir
to the next at every time step. The fraction of water moved is written on top of the edge.
Figure 1: Pump system
We want to prove the following theorem. We will do this step by step.
Theorem: Consider a system consisting of k reservoirs such that the entries of each column in the system's
state transition matrix sum to one. If s is the total amount of water in the system at timestep n, then total
amount of water at timestep n +1 will also be s.
(a) Rewrite the theorem statement for a graph with only two reservoirs.
(b) Since the problem does not specify the transition matrix, let us consider the transition matrix A =
a11 a12 and the state vector x[n]
[a21 a22]
the theorem statement in mathematical form.
Hint: In general, it is helpful to write as much out mathematically as you can in proofs. It can also be
helpful to draw the transition graph.
Write out what is “known" or what is given to you in
[x2[m].
(c) Now write out what is to be proved in mathematical form.
(d) Prove the statement for the case of two reservoirs.
(e) Now use what you learned to generalize to the case of k reservoirs. Hint: Think about A in terms of
its columns, since you have information about the columns.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda4a7445-4cc9-4690-867b-b0862e018c2b%2F5551b06a-00db-45fb-81bc-9d80b9d4f7f3%2Fztwnll9_processed.png&w=3840&q=75)
Transcribed Image Text:3. Properties of Pump Systems - II
Learning Objectives: This problem builds on the pump examples we have been doing, but is meant to help
you learn to do proofs in a step by step fashion. Can you generalize intuition from a simple example?
We consider a system of reservoirs connected to each other through pumps. An example system is shown
below in Figure 1, represented as a graph. Each node in the graph is marked with a letter and represents a
reservoir. Each edge in the graph represents a pump which moves a fraction of the water from one reservoir
to the next at every time step. The fraction of water moved is written on top of the edge.
Figure 1: Pump system
We want to prove the following theorem. We will do this step by step.
Theorem: Consider a system consisting of k reservoirs such that the entries of each column in the system's
state transition matrix sum to one. If s is the total amount of water in the system at timestep n, then total
amount of water at timestep n +1 will also be s.
(a) Rewrite the theorem statement for a graph with only two reservoirs.
(b) Since the problem does not specify the transition matrix, let us consider the transition matrix A =
a11 a12 and the state vector x[n]
[a21 a22]
the theorem statement in mathematical form.
Hint: In general, it is helpful to write as much out mathematically as you can in proofs. It can also be
helpful to draw the transition graph.
Write out what is “known" or what is given to you in
[x2[m].
(c) Now write out what is to be proved in mathematical form.
(d) Prove the statement for the case of two reservoirs.
(e) Now use what you learned to generalize to the case of k reservoirs. Hint: Think about A in terms of
its columns, since you have information about the columns.
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