Let k be a fixed positive integer and let G = (V, E) be a loop-free undirected graph, where $\operatorname{deg}(\mathrm{v}) \geq \mathrm{k}$ for all $\mathrm{v} \in \mathrm{V}.$ Prove that G contains a path of length k.

Suppose you are looking along a line through the centers of two circular (but separate) wire loops, one behind the other. A battery is suddenly connected to the front loop, establishing a clockwise current. $(e)$ Is there a force between the two loops?

Exercises refer to the following algorithm segment. For each positive integer n, let

$$a_n$$

be the number of iterations of the while loop. while (n>0), n :=n div 2, end while. Trace the action of this algorithm segment on n when the initial value of n is 27.

The pilot of an airplane executes a vertical loop which in part follows the path of a cardioid, $r=600(1+\cos \theta) \mathrm{ft}$, where $\theta$ is measured in clockwise direction from the vertical axis. If his speed at $A\left(\theta=0^{\circ}\right)$ is a constant $v_P=80 \mathrm{ft} / \mathrm{s}$, determine the vertical force the seat belt must exert on him to hold him to his seat when the plane is upside down at $A$. He weighs $150 \mathrm{lb}$.

A single loop of wire is placed in a time-varying magnetic field, and an induced emf of $3.5 \mathrm{~V}$ is observed. Suppose $25$ identical loops with the same orientation are now connected in series. What is the value of the total induced emf?

A principal node is a. a closed path or loop where the algebraic sum of the voltages must equal zero. b. the simplest possible closed path around a circuit. c. a junction where branch currents can combine or divide. d. none of the above.

Write a program with a loop that lets the user enter a series of integers. The user should enter −99 to signal the end of the series. After all the numbers have been entered, the program should display the largest and smallest numbers entered.

A multiple-loop model of an urban ecological system might include the following variables: number of people in the city $(P)$, modernization $(M)$, migration into the city $(C)$, sanitation facilities $(S)$, number of diseases $(D)$, bacteria/area $(B)$, and amount of garbage/area $(G)$, where the symbol for the variable is given in parentheses. The following causal loops are hypothesized:

1. $P \rightarrow G \rightarrow B \rightarrow D \rightarrow P$
2. $P \rightarrow M \rightarrow C \rightarrow P$
3. $P \rightarrow M \rightarrow S \rightarrow D \rightarrow P$
4. $P \rightarrow M \rightarrow S \rightarrow B \rightarrow D \rightarrow P$ Sketch a signal-flow graph for these causal relationships, using appropriate gain symbols. Indicate whether you believe each gain transmission is positive or negative. For example, the causal link $S$ to $B$ is negative because improved sanitation facilities lead to reduced bacteria/area. Which of the four loops are positive feedback loops and which are negative feedback loops?

The nested loops for (int i = 1; i <= height; i++) { for (int j = 1; j <= width; j++) { cout << "*"; } cout << endl; } display a rectangle of a given width and height, such as

****
****
****

Write a single for loop that displays the same rectangle.

Suppose the magnetic field of the preceding problem oscillates with time according to $B = B _ { 0 } \sin \omega t$. What then is the emf induced in the loop when its trailing side is a distance d from the right edge of the magnetic field region?

Write a program with a loop that asks the user to enter a series of positive numbers. The user should enter a negative number to signal the end of the series. After all the positive numbers have been entered, the program should display their sum.

In a program you need to store the populations of $12$ countries.

a. Define two arrays that may be used in parallel to store the names of the countries and their populations.

b. Write a loop that uses these arrays to print each country’s name and its population.n 3(a).

The feedback current amplifier in Fig. P10.26 utilizes an op amp with an input differential resistance $R_{i d}$, an open-loop gain $\mu$, and an output resistance $r_o$. The output current $I_o$ that is delivered to the load resistance $R_L$ is sensed by the feedback network composed of the two resistances $R_M$ and $R_F$, and a fraction $I_f$ is fed back to the amplifier input node. Find expressions for $A \equiv I_o / I_i$,$\beta \equiv I_f / I_o$, and $A_f \equiv I_o / I_s$, assuming that the feedback causes the voltage at the input node to be near ground. If the loop gain is large, what does the closed-loop current gain become? State precisely the condition under which this is obtained. For $\mu=10^4 \mathrm{~V} / \mathrm{V}, R_{\text {id }}=1 \mathrm{M} \Omega, r_o=100 \Omega$, $R_L=10 \mathrm{k} \Omega, R_M=100 \Omega$, and $R_F=10 \mathrm{k} \Omega$, find $A$, $\beta$, and $A_f$.

Choose a word or words from the box below to complete the following sentence.

Fish ________________ systems pump blood in a single loop through a heart with two main chambers.