Discuss the significance of the three-pillory scenes in Hawthorne's The Scarlet Letter?

The platform of the pillory, or the scaffold, is an actual
platform that a guilty person must stand on for a length of time in order to be put on
display for their sins and suffer public humiliation.

1st
scene:

who: Hester and (baby)
Pearl. 
when:
Noon
reason: Hester's punishment as dictated by
the laws of the colony for comitting adultery
significant
spectators: the entire colony, the magistrates and ministers,
Chillingworth (who has just arrived)
outcome:
Hester will not reveal who the father of the baby is, even when persuaded by Dimmesdale
himself (sit. irony); Hester shows no signs of remorse nor fear - looks very
proud.

2nd
scene:

who: Dimmesdale then
Hester and Pearl with him
when: middle of the
night about 7 years later
reasons: Dimmesdale
seeks resolution and forgiveness for hiding his secret sin; in this symbolic act of
repentance (in front of no one) he believes this will provide his soul with
relief
significant spectators: no one, but
Chillingworth shows up at the very end and it is a mystery how long he's been there and
how much he has seen
outcome: Pearl and Hester
join Dimmesdale and for a moment he has an unusual feeling of strength; Pearl asks him,
"Will you stand with us, we three together, tommorrow noon-tide?"  She continues to
condemn him for keeping the secret; a shooting star then bursts through the sky leaving
what looks like a large letter "A" behind it; the next morning the townspeople who saw
it say it was an A for "angel" as the great Gov. Winthrop died the night before; also,
Dimmesdale forgets his glove on the steps, and when it is found the next morning the
townspeople say that Satan himself must have taken it and placed it
there.

3rd
scene:

who: Dimmesdale, Pearl,
Hester
when: middle of the day, just after the
annual election day speech made by
Dimmesdale
reason: Hester and Dimmesdale plan to
board a boat and leave for England; this is Dimmesdale's final action to absolve himself
of his sin; he climbs the steps first then encourages Hester and Pearl to join
him
significant spectators: the entire colony
plus a bunch of visitors
outcome: Dimmesdale rips
his shirt off and although it is not said exactly what is there, it is assumed he has
his own scarlet letter on his chest; the entire colony finally
knows the truth of who the father is; Chillingworth approaches the steps and tries to
get Dimmesdale to come down, then says, basically, this is the one and only place you
could have escaped me - confirming that the only way for Dimmesdale to truly repent was
to come clean before the people; oh yes, and Dimmesdale dies.

How does the short story "Babylon Revisited" demonstrate the universal truth of sacrifice?

Here is one definition of
sacrifice:

. .
. the act of deliberately following a course of action that has a high risk or certainty
of suffering, personal loss or death (which could otherwise be avoided), in order to
achieve a perceived benefit for self or
others.

In "Babylon
Revisited," Charlie Wales is clearly a flawed man, but he is also a man who chooses
self-sacrifice at significant times in his life. Although he loves his daughter Honoria
deeply, Charlie gives up custody of the child, knowing that he cannot care for her and
that she will have a stable life in Paris with his wife's sister, Marion Peters, and her
husband.

After battling and apparently overcoming his
alcoholism, Charlie moves to Prague where he works responsibly and rebuilds the fortune
he had lost. He is safe in Prague, living a healthy life far removed from the excesses
of his old, decadent life in Paris. His love for Honoria, however, compels him to
sacrifice the safety of his new environment in Prague and return to Paris, the place
where old memories of his painful past await him. Charlie sacrifices peace, as well, by
returning to Paris to regain custody of Honoria. Familiar sights and sounds remind him
of the many ways he had destroyed his life; he is haunted by dreams of Helen, his dead
wife. Returning to Paris exposes Charlie to great emotional pain and threatens to wreck
his sobriety; he knows the dangers to his hard won and tenuous well being, but he goes.
He risks all for love of Honoria.

In facing Marion Peters
again, Charlie sacrifices his pride in order to regain his daughter. He appears humble
before her, making only weak efforts to defend himself against her contempt and
disparaging remarks. When Marion decides that she will not return Honoria to him at that
time, Charlie accepts her decision. Fighting Marion for custody of his daughter would
surely have disrupted Honoria's life and destroyed her security. Charlie chooses instead
to leave his daughter behind, for a while longer. He will endure the separation for her
benefit. Charlie sacrifices what is most dear to him out of his love for his
daughter.

How do we derive the expansion for sin 3x?

I think by expansion you mean sin 3x in terms of sin
x.

We start with the relations sin (x + y) = sin x* cos y +
cos x*sin y and cos (x + y) = cos x* cos y – sin x*sin
y

sin 3x = sin (2x + x) = sin 2x* cos x + cos 2x*sin
x

=> sin (x + x)* cos x + cos (x +x)*sin
x

=> [sin x* cos x + cos x*sin x]* cos x + [cos x*
cos x – sin x*sin x]*sin x

=> 2*sin x * (cos x) ^2 +
(cos x) ^2 * sin x – (sin x) ^3

=> 3*sin x (cos x)
^2 – (sin x) ^3

now (cos x) ^2 + (sin x) ^2 = 1 or (cos x)
^2 = 1 – (sin x) ^2

=> 3* sin x*(1 – (sin x) ^2) -
(sin x) ^3

=> 3* sin x – 3(sin x) ^2) - (sin x)
^3

=> 3*sin x – 4*(sin x)
^3

Therefore we get sin 3x = 3*sin x – 4*(sin
x) ^3

what would happen if there was no carbon cycle/ the carbon cycle stopped?

All living things contain carbon. Remember the equation
for cellular respiration--oxygen + carbohydrates--->carbon dioxide + water (and
releases energy). The reverse, photosynthesis, is the way that energy is captured as
sunlight, and made available for living things. Both carbohydrates and carbon dioxide
contain
carbon.

Carbon is
also found in various other compounds in our bodies, such as proteins, lipids, and
genetic material. It helps make up our teeth and bones. When carbon cycles, it also does
so through non-living things. The amount of carbon does not change; we can't get more.
If carbon did not cycle through various forms, there would not be any available for
living things, and life would not be possible. At least, not any type we are used
to.

If sin y = cos 3y, determine an exact value for angle y.

We have to find y such that sin y = cos
3y.

We know that sin(y) = cos(pi/2-y),so we can convert the
given equation to:

sin(y) = cos(pi/2-y) =
cos(3y)

We now equate pi/2 - y and
3y

pi/2 - y = 3y

=>
pi/2 = 3y + y

=> 4y =
pi/2

=> y =
pi/8

Therefore y = pi/8 +
2*n*pi

Prove the identity 1+3+5+...+(2n-1)=n^2

For demonstrating the identity, we'll use the method of
mathematical induction, which consists in 3
steps:

1) verify that the method works for the number
1;

2) assume that the method works for an arbitrary number,
k;

3) prove that if the method works for an arbitrary
number k, then it work for the number k+1, too.

4) after
the 3 steps were completed, then the formula works for any
number.

Now, we'll start the first
step:

1) 1=1^2 => 1=1
true.

2) 1+3+5+...+(2k-1)=k^2 ,
true.

3) If 1+3+5+...+(2k-1)=k^2,
then

1+3+5+...+(2k-1)+(2k+2-1)=(k+1)^2

Let's
see if it is true.

For the beginning, we notice that the
sum from the left contains the assumed true equality, 1+3+5+...+(2k-1)=k^2. So, we'll
re-write the sum by substituting 1+3+5+...+(2k-1) with
k^2.

k^2 + (2k+1) =
(k+1)^2

We'll open the
brackets:

k^2 + 2k+1 = k^2 + 2k+1
true.

4) The 3 steps were completed, so the identity is tru
for any value of
n.

1+3+5+...+(2n-1)=n^2

How does Hamlet's father manipulate Hamlet even when he is dead?

The fact that Prince Hamlet's father returns from the dead
as a ghost to tell his son what happened to him is the basis of the entire play.  Hamlet
respected his father in life, and wants to do right by him in death, and that is what is
central to most of the actions of Hamlet for the rest of the
play. 

Even before the ghost speaks he is manipulating
people.  He is appearing to guards and friends of Hamlet, but not speaking so as to get
them to tell Hamlet about this strange thing -- a ghost.  By having his friends all say
they saw a ghost will make Hamlet more likely to believe what he sees and not think it
is merely a hallucination caused by grief.

Once the ghost
is actually talking to Hamlet he says several things to manipulate Hamlet into
completing his request for vengeance.  First of all he starts talking, in kind of scary
generalities about where he is in the afterlife -- it is purgatory where he is "doomed
for a certain term to walk the night, And for day confined to fast in fires."  This
creates a certain amount of sympathy in Hamlet for the horrible afterlife his father is
suffering.  After that is established, he kind of draws out the suspense of his
message.  First he mentions that he was murdered, then he tells how, and finally
he delivers the fateful news, that "the serpent who did sting thy father's life now
wears his crown."  By building up the story this way it is manipulating Hamlet's
attention and making the last note the most powerful and
shocking.

The next phase of manipulation goes on for
several lines when the ghost provides the more specific and gross details of his death,
thus manipulating Hamlet's sympathy and anger.  He also makes several negative
statements about Claudius's seduction of Gertrude and how Claudius is such a lesser man
than he.  This feeds right into how Hamlet already feels about the gross, incestuous
marriage of his mother to Claudius and how he has already expressed his displeasure at
the kind of man Claudius is.  In a sense, Hamlet is very prepared to hear awful things
about Claudius -- he thought those very things even before he learned that Claudius
murdered his father.

The ghost also manipluates Hamlet by
drawing a sympathetic picture of Gertrude so that Hamlet won't try to punish her for her
action in marrying Claudius.  Claudius is the only one that the ghost wants revenge on. 
He tells Hamlet to leave his mother to heaven's
judgment.

His final manipulation is to the command to
"remember me."  He wants Hamlet to have at the forefront of his thoughts the father that
Hamlet used to be and the facts that he learned today.  Together, all of that should
keep Hamlet highly motivated to avenge his father's death by taking care of Claudius
once and for all.