How did Aram justify Mourad's act of stealing the white horse in "The Summer of the Beautiful White Horse?"

To understand this question, take a look at Aram's thoughts and feelings when he is first offered the chance to ride the horse by his cousin, Mourad:

Well, it seemed to me stealing a horse for a ride was not the same thing as stealing something else, such as money. For all I knew, maybe it wasn't stealing at all. If you were crazy about horses the way my cousin Mourad and I were, it wasn't stealing.

In Aram's mind, when Mourad took this horse, he had no intention of selling it to make a profit and, therefore, his actions cannot be described as theft. As such, he does not need to justify what has happened because he does not believe that his cousin, Mourad, has broken the law.
Moreover, Aram is easily able to justify his own riding of the horse. He allows himself to ride because he is "crazy" about horses and this provides him with a rare opportunity to indulge his passion. His family is so poor that this chance might not come about again so he must seize it, before it is time to return the horse to its rightful owner.

In William Saroyan’s short story “The Summer of the Beautiful White Horse,” Aram justifies his cousin’s acquisition of the horse using his personal reasoning process. By examining the text for evidence, the reader can follow Aram’s thought process.
Aram and Mourad come from a poor but proud, honest, self-sufficient Armenian family. Stealing is unheard of in their clan. When Mourad arrives with the horse in the middle of night, Aram is forced to justify how the animal came to be in his possession. After thinking about it, Aram decides his cousin did not steal the horse because he was not going to sell it for money. The boys were only going to ride it, which he reasoned did not constitute stealing. If they decided to sell it, which would never enter their minds, then the line would be crossed, and the animal would be considered stolen goods.

Aram justifies Mourad's stealing of the horse for two reasons; one, Aram does not believe it is the same as stealing, as they weren't doing it for money but to ride the horse for fun. He also justifies it because he has an enthusiasm and love for horses, and this opportunity may not come again, since their family is so poor.

In Macbeth, what is important about the witches' prophecy?

The witches' first prophecy is what causes Macbeth to think that he could be king to begin with. There is no indication that he had that ambition before they spoke to him. The witches demonstrate that they can tell the future by predicting that he will become Thane of Cawdor, which immediately takes place. This experience makes Macbeth believe that he is fated to become the king, but, of course, he is not entirely content just to let that happen without acting. With some spurring from Lady Macbeth, he murders Duncan. But that is not enough for Macbeth; the witches also predicted that Banquo's children would be kings, implying that at some point Macbeth's line would die out and Banquo's line would rule instead. Macbeth then has to have Banquo killed too.
The witches prompt every murder that Macbeth commits or commissions following the murder of King Duncan. He visits the witches again in act 4, scene 1. They tell him to "beware the Thane of Fife! Beware MacDuff!" This prophecy results in Macbeth having MacDuff's family murdered when he learns that MacDuff has fled to England, out of his reach for the present. The witches' prophecies make Macbeth commit multiple murders in order to try to secure his future power and safety. However, they also cause him to believe that he is secure, telling him that "none of woman born can harm Macbeth." This prophecy turns out to be a trick because MacDuff was born by C-section, rather than through labor, so he does not count as being "born of woman" and can kill Macbeth. The witches play with both Macbeth's ambitions and his fears at once.
The witches' prophecies are important because without them, Macbeth would have had no reason to kill Duncan. Planting the idea in Macbeth's head is what makes the events of the play happen: no prophecies, no play. They also spur the plot onward once it is in motion, prompting more and more killing until Macbeth is finally utterly destroyed.

The weird sisters' prophecy has several important functions in the play. First, their meeting on the blasted heath creates an ominous atmosphere of supernatural horror at the start of the play. It is one of the most dramatic scenes in the play and an audience favorite. 
More importantly, Shakespeare lived in a place and period where most of his audience would have been Christian and believed witches to be evil and allied with the Devil. Thus they provide a religious and moral context for understanding the play.
Next, they set in motion the main plot of the play. It is their prophecy that causes Macbeth to turn his ambition, which has previously been manifested in loyal service to Duncan, to focusing on killing Duncan and taking the throne himself. Because it is evil witches who tempt him to do this, the audience knows that his choices are not justified and that he will prove evil as well. The witches constantly bring out the worst in Macbeth.

sum_(n=1)^oo n*e^(-n/2) Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

Integral test is applicable if f is positive and decreasing function on interval [k,oo) where a_n = f(x).  
If int_k^oo f(x) dx is convergent then the series sum_(n=k)^oo a_n is also convergent.
If int_k^oo f(x) dx is divergent then the series sum_(n=k)^oo a_n is also divergent.
For the  series sum_(n=1)^oo n*e^(-n/2) , we have a_n =n*e^(-n/2) then we may let the function:
f(x) =x*e^(-x/2) .
 The graph of f(x) is:

 As shown on the graph, f(x) is positive on the interval [1,oo) .  Based on the behavior of the graph as x increases, the function eventually decreases. We can confirm this by applying First Derivative test.  To determine the derivative of the function, we may apply the Product rule for differentiation: d/(dx) (u*v)= v* du+ u *dv .
Let: u =x then du = 1
       v=e^(-x/2)  then dv =- e^(-x/2)/2
Note:  d/(dx)e^(-x/2) = e^(-x/2) * d/(dx) (-x/2)
                      =e^(-x/2) *(-1/2)
                      =- e^(-x/2)/2
Applying the Product rule for differentiation, we get:
f'(x) =e^(-x/2) * 1 + x *- e^(-x/2)/2
         =e^(-x/2) - (xe^(-x/2))/2
         = (e^(-x/2) (2-x))/2
Solve for critical values of x by applying f'(x) =0 .
(e^(-x/2) (2-x))/2 =0
(e^(-x/2) (2-x))=0
 Apply zero-factor property:
(2-x)=0 then x=2
Using test point x=5 after x=2 , we get:
f'(5) = (e^(-5/2) (2-5))/2 ~~ -0.12313 .
When f'(x) lt0 , then the function is decreasing for the given integral.
Then f(x)=x*e^(-x/2) from the interval [2, oo) . Since the function is ultimately decreasing on the interval [1,oo) we may apply the integral test:
int_1^oo x*e^(-x/2) dx= lim_(n-gtoo) int_1^tx*e^(-x/2)dx
To determine the indefinite integral of int_1^t x*e^(-x/2)dx , we may apply u-substitution by letting:  u =-x/2 or x=-2u then du = -1/2 dx or -2du =dx .
The integral becomes:
int x*e^(-x/2)dx=int (-2u)*e^u*(-2du)
                      = int 4ue^u du
                      = 4 int ue^udu
Apply the integration formula for exponential functions: int xe^xdx=(x-1)e^x+C.
4 int ue^udu=4 *(u-1)e^u
                     = 4ue^u -4e^u
Plug-in u =-x/2 on 4ue^u -4e^u , we get:
int_1^t x*e^(-x/2)dx =4(-x/2)e^(-x/2) -4e^(-x/2)|_1^t
                     =-2xe^(-x/2) -4e^(-x/2)|_1^t
Applying definite integral formula: F(x)|_a^b = F(b)-F(a).
-2e^(-x/2) -4e^(-x/2)|_1^t=[-2te^(-t/2) -4e^(-t/2)]-[-2*1e^(-1/2) -4e^(-1/2)]
                             =-2te^(-t/2) -4e^(-t/2)+2e^(-1/2) +4e^(-1/2)
                             =-2te^(-t/2) -4e^(-t/2)+6e^(-1/2)
Applying int_1^t x*e^(-x/2)dx =-2te^(-t/2) -4e^(-t/2)+6e^(-1/2) , we get:
lim_(n-gtoo) int_1^tx*e^(-x/2)dx =lim_(n-gtoo)[-2te^(-t/2) -4e^(-t/2)+6e^(-1/2)]
                              =lim_(n-gtoo)-2te^(-t/2) -lim_(n-gtoo)4e^(-t/2)+lim_(n-gtoo)6e^(-1/2)
                              =-2*ooe^(-oo) -4e^(-oo)+6e^(-1/2)
                             =0-0+6/e^(1/2)
                            =6/e^(1/2) or 6/sqrt(e)
The  lim_(n-gtoo) int_1^tx*e^(-x/2)dx =6/sqrt(e) implies that the integral converges.
Conclusion:
The integral int_1^oo x*e^(-x/2)dx is convergent therefore the series sum_(n=1)^oo n*e^(-n/2) must also be convergent.

How did geographic isolation and cultural diffusion shape society in ancient Egypt?

Geographically, ancient Egypt was an isolated region. There was a big desert to its west, what is now called the Libyan Desert. Its north was surrounded by the Mediterranean Sea and its east by the Red Sea. The Kush Kingdom covered Egypt’s southern border. As such, there were only two points of entry by land into Egypt: the south or the northeast. Most of Egypt’s enemies entered it through the northeastern route via Palestine and across the Sinai Peninsula. To protect itself against these invaders, Egypt built fortresses along the Sinai Peninsula. The surrounding deserts and oceans also acted as natural borders that kept out intruders.
The Egyptians and their southern neighbors from the Kingdom of Kush interacted a lot. In fact, they even ruled each other at certain points in history. They were also great trading partners. Evidence of the cultural diffusion between these two kingdoms exists in similarities in religion and burial rites. The Kushites buried their dead in pyramids that resembled the Egyptian pyramids. Both cultures worshiped the gods Osiris, Thoth, and Isis.
http://www.angelfire.com/nt/theology/06egypt.html

How would you persuade a courtroom that Friar Laurence is to blame for Romeo and Juliet's deaths?

To persuade a jury that Friar Laurence was responsible for the deaths of Romeo and Juliet, it would be important to establish the following:
First, Friar Laurence was the responsible adult who should have found a tactful way to inform the Capulet and Montague parents of the young lovers' determination to get married. If I were the lawyer, I would emphasize the innocence of both Romeo and Juliet. I would show that they were teenagers who didn't understand the risks they were taking in marrying and didn't understand they were rushing into matrimony completely unprepared. A wiser friar would have known that it is insanity to marry two young people who had only met a day ago. Why was Friar Laurence enabling this? Why wasn't he acting like a grown-up?
Second, I would argue that his marrying them led directly to their deaths because it forced Juliet to take desperate measures to avoid marrying Paris. Why on earth was the friar urging her to take a potion mimicking death? As she made clear in her soliloquy, she was terrified to do so and didn't want to be lying in a crypt surrounded by dead relatives. Why didn't he, rather than suggesting a risky plot, urge her to go to her parents and let them know what she had done? If they had known she was already married, how could they have forced her to marry Paris?
I would further argue that Friar Laurence was willing to allow the young lovers to engage in a high-risk stratagem for the selfish reason of covering up his role in marrying them. I would also argue that anyone with half a grain of sense would know how impulsive Romeo was and take care not to get him mixed up in an ill-conceived plan that so easily could go wrong. Friar Laurence sacrificed the naive and trusting young people to save himself.

What did Gerda Lerner seem to be saying about women as victims of oppression? What do women sacrifice by accepting this?

In this essay, published originally in 1975, Lerner, a feminist and a historian, is surveying various trends in the study of women's history, discussing their benefits, but also, more importantly, their drawbacks. Fundamentally, she is suggesting that the ways women's history has been studied up to that point have tended to operate, sometimes unintentionally, within the same patriarchal assumptions that oppressed women in the first place. So in studies that have, quite accurately, depicted various ways in which women have been historically oppressed (the subject of your question), historians have tended to obscure or downplay the ways in which women resisted oppression. This makes it appear, in Lerner's words, that "women were largely passive," or that their actions were simply reactions to "male pressures or to the restraints of patriarchal society." Thus women as historical actors are placed "in a male-defined conceptual framework: oppressed, victimized by standards and values established by men." To use the word that social historians often use to describe the actions of the historical people they study, this approach obscures the agency of historical women. "The true story," Lerner says, of these women is "of their ongoing functioning in that male-defined world, on their own terms," and because focusing simply on oppression does not reveal this, it is "of limited usefulness to the historian." So the focus on oppression is, like a focus on "great" women, for example, always going to yield a very simplistic picture of women's historical lives even as it draws much needed attention to the systems of oppression faced by women. This is a very common concern among historians, not just concerning women, but also in relation to enslaved people, Native Americans, working-class people, and many others. How do we illustrate the structures and processes that oppress people without overlooking the ways that those same people figure out how to live within them?
http://reganhistory.weebly.com/uploads/2/5/5/4/25540442/women_and_themaking_of_america.pdf

How can I write a character sketch of the Queen of Brobdingnag?

To write a character sketch, think about how the descriptions of the queen paint a portrait of her and reveal her personality. For example, the Queen of Brobdingnag is a kind though somewhat silly person who treats Gulliver, who is much smaller than the inhabitants of her land, as a kind of pet. When she first encounters Gulliver, she is delighted with him and asks his master if she can purchase Gulliver from him. Gulliver tells her the story of his mistreatment at the hands of his former master and the queen is "surprised at so much wit and good sense in so diminutive an animal." She brings him to the king, who at first treats Gulliver dismissively until the queen, with her intelligence and wit, commands Gulliver to give an account of himself. The queen kindly instructs a carpenter to construct a bedchamber for Gulliver and orders clothes to be made for him out of the finest silk.
The queen, who also keeps a dwarf as a kind of pet, grows so fond of Gulliver that she cannot dine without him, and she keeps him by her elbow at the table. However, she does not really understand his fears at being so little, and she does not understand why he is so bothered by giant flies. The queen also constructs a rowboat and trough for Gulliver, in which she and members of her court enjoy watching him row. She is amused by Gulliver and is generous, but she is not particularly deep or insightful, as she doesn't understand Gulliver's struggles as a tiny person in her kingdom. 

A character sketch outlines the nature of a character in a work of literature. Any assertions made about the character must be illustrated with quotes (or paraphrases) from the work to support your argument. It's important to include quotes to show that this is not your subjective opinion of the character but a text-based and objective argument. 
That being said, this text doesn't offer that much information about the Queen of Brobdingnag, which means we need to focus closely on what Swift does tell us while  keeping in mind that this is a satire in which events are being filtered through the mind of the often clueless Gulliver. How does Gulliver initially react to the Queen's large size? How do his impressions change and what does that say about her and also the prejudices we hold based on appearance? Why do you think she is one of the only female characters in the work that Gulliver pays any real attention to? What does her reaction to her dwarf's mistreatment of Gulliver say about her? Why does Gulliver think she is witty? As you address these questions and gather more insights about the Queen, be sure to find quotes to back yourself up.