In Hamlet, how does Claudius manipulate Gertrude, Rosencrantz and Guildenstern, Hamlet, and Laertes to achieve his own ends?

Concerning Shakespeare's Hamlet, in
short, King Claudius manipulates the characters you ask about in the following
ways:

• He gets Gertrude to marry him and, thus,
  gets the crown.  He also manages to keep Gertrude viewing Hamlet on his terms and from
  his point of view for most of the play:  at least until the bed chamber scene in Act
  3.4, and possibly until the final scene in the play when she realizes Claudius has
  poisoned the cup meant for Hamlet.  Gertrude is constantly on Claudius's side and
  perceives every issue that concerns Hamlet from the same point of view as Claudius (that
  Hamlet should shake off his sadness, that he should stay at Elsinore and not go back to
  school, that Hamlet may be "mad" because he is in love with Ophelia). 
  


• Claudius gets Ros. and Guil. to spy on Hamlet for him. 
  He turns them from being Hamlet's friends to his own agents.  He summons them to
  Elsinore for this purpose and uses them for it.  He repeatedly sends them on errands to
  Hamlet:  to find out what's bothering him, for example, and to find out where Polonius's
  body is. 


• Claudius redirects Laertes's wrath over the
  death of Polonius away from himself and toward Hamlet.  He leads Laertes to think
  that he is helping him kill Hamlet for Laertes's sake, when in reality he is helping
  Laertes in order to protect his own
  crown. 

Concerning Hamlet, however, Claudius
attempts to manipulate Hamlet, but it doesn't work out in his favor.  Claudius thinks he
is playing a cat-and-mouse game with Hamlet, and he is--the only trouble is that Hamlet
is the cat, not Claudius. 

Notice how much time and effort
Claudius spends trying to figure out why Hamlet is "mad."  This is by design--Hamlet's
design.  He tells Horatio that he will be putting on an "antic disposition" (Act
1.5.171)--pretending to be mad.  This is a diversion designed by Hamlet.  And it works. 
Claudius spends all his time trying to figure out why Hamlet is insane, and never
figures out--until Hamlet wants him to--that Hamlet knows about Claudius's murder of
King Hamlet. 

Only when Hamlet gets what he considers to be
proof or confirmation that the Ghost is telling him the truth and that Claudius is
guilty (by watching Claudius's reaction to the murder scene in the play-within-the-play)
does Claudius know Hamlet knows.  And notice what Hamlet tells Claudius the name of the
play is--The Mousetrap.  Claudius doesn't manipulate Hamlet; Hamlet manipulates
Claudius.  Hamlet is the cat.

What is the centerpiece on the table at dinner?

The centerpiece on the table at dinner consists of ten
china figures of indians.  As each of the guests meets their fate, a china figure
mysteriously disappears from the centerpiece.  After the death of the first guest, Tony
Marston, Vera makes the connection between the china figures, the verse from "The Ten
Little Indians" poem, and the manner in which Marston died. After the second death, Mrs.
Rogers, it is discovered that there are only eight china figures remaining in the
centerpiece.  This pattern continues as one by one, the guests meet their untimely
end.

What is the first term of arithmetic sequence if the sum a1+a2+...a13=130? a4,a10,a7 are also consecutive terms of geometric sequence

We'll write the sum of the first 13th terms of an
arithmetical sequence:

S13 =
(a1+a13)*13/2

130 =
(a1+a13)*13/2

We'll divide by 13 both
sides:

10 = (a1+a13)/2

a1 +
a13 = 20

But a13 = a1 + 12d, where d is the common
difference of the arithmetical sequence.

a1 + a1 + 12d =
20

2a1 + 12d = 20

a1 + 6d = 10
(1)

We also know that a4,a10,a7 are the consecutive terms
of geometric sequence.

a10^2 =
a4*a7

We'll write a4,a10,a7 with respect to a1 and the
common difference d.

(a1 + 9d)^2 = (a1 + 3d)(a1 +
6d)

We'll expand the square and we'll remove the
brackets:

a1^2 + 18a1*d + 81d^2 = a1^2 + 9a1*d +
18d^2

We'll eliminate a1^2, we'll move all terms to one
side and we'll combine like terms:

9a1*d + 63d^@ =
0

We'll divide by 9:

a1*d +
7d^2 = 0

We'll factorize by
d:

d(a1 + 7d) = 0

a1 = -7d
(2)

We'll substitute (2) in
(1):

-7d + 6d = 10

-d =
10

d = -10

a1 = 10 -
6d

a1 = 10 + 60

a1 =
70

The first term of the arithmetical
progression is a1 = 70, for d = -10 and a1 = 10 for d =
0.

What is J.R. Oppenheimer's importance in US history?

Oppenheimer was not only chief scientist on
the atomic bomb project but was the first major atomic scientist post-war to take a
public stand against continued research into nuclear weapons, specifically the
development of the hydrogen bomb. The project was originally termed the
"Manhattan Engineering District," although the code name changed monthly during the
war.  It was a joint project between the US, UK and Canada, carried out in the US
because there was no secure location in Britain during the war.  The British weapons
project TUBE ALLOYS was the basis of the work.  After the war the scientists involved
split into groups, some opposing further weapons research, some for it.  Edward Teller
was a brilliant physicist who was also an opportunist, and seeing an unlimited future of
government sponsored research ahead he tirelessly worked to ruin Oppenheimer's
reputation.  In the Red Scare in the wake of the disastrous beginning of the Korean War,
Oppenheimer was easy prey for people like Teller and McCarthy.  In 1953 his security
clearances were revoked,effectively ending his work with governmental projects.  From
1947 through 1966 he was director of the Institute for Advanced Studies, at Princeton. 
Prior to the war he had been simultaneously assistant professor of physics at UC Berkely
and the California Institute of Technology.

Interestingly
enough, Robert Oppenheimer in the 1930s was the first physicist to suggest the existence
of what are today termed "black holes," as well as important work in the theory of
cosmic ray showers and what eventually became known as quantum tunneling.  He was one of
the greatest of American scientists, and a man who the country owed a great debt of
gratitude to but who was treated shabbily.

In Monster, why is it said that it is best to cry at night while someone is screaming?

Steve states that the best time to cry is at night
because, normally, someone is being attacked and if the noise of the attack is loud
enough no one would be able to hear you cry.  Steve is fearful that if someone hears you
cry that you will be seen as weak and that you will be the next one to get attacked. 
Steve knows that jail/prison is a place full of violence, that he does not being here,
and showing any weakness will cause him problems with the other
inmates.

What is the equation of the perpendicular bisector of the line between (6, 3) and (3, 4)?

Since the line is the bisectrix of the segment whose
endpoints are (6, 3) and (3, 4), the midpoint of the segment is on this
line.

We'll calculate the
midpoint:

x mid = (6+3)/2

x
mid = 9/2

y mid = (3+4)/2

y
mid = 7/2

Since the bisectrix is perpendicular on the
segment, the product of the slopes of the perpendicular lines is
-1.

First, we'll write the equation of the
segment:

(6-3)/(x-3) =
(3-4)/(y-4)

3/(x-3) =
-1/(y-4)

-x + 3 = 3y -
12

We'll put the equation in the slope intercept
form:

3y = -x + 15

y = -x/3 +
5

The slope of the segment line is m1 =
-1/3

m2*m1 = -1

m2 =
-1/m1

m2 = 3

The equation of
the perpendicular line is:

y - ymid = m2(x -
xmid)

y - 7/2 = 3(x - 9/2)

(2y
- 7)/2 = (6x - 27)/2

We'll simplify and we'll
get:

2y - 7 = 6x - 27

The
equation of the perpendicular bisectrix line is 6x - 2y - 20 =
0.

We'll simplify and we'll
get:

3x - y - 10 =
0

In Shakespeare's Macbeth, why is Act 2 Scene2 a turning point?Focus on how Macbeth and Lady Macbeth are portrayed in this scene.

In Shakespeare's Macbeth, Act II,
scene ii, the turning point arises with regard to the change of positions of Macbeth and
Lady Macbeth.

At the start of the play, Macbeth is a great
and celebrated warrior. He is strong and self-assured. His wife is a schemer who wants
to be queen, nagging her husband to take what he wants, regardless of what he must do to
get it. However, after Macbeth murders Duncan, his King, his friend, and his cousin, he
is undone. He comes back to their rooms carrying the incriminating daggers. He has heard
the king's servants awake and say their prayers, but when they said, "Amen," Macbeth was
unable to utter the word. He says:

readability="7">

But wherefore could not I pronounce 'Amen'? / I
had most need of blessing, and 'Amen' / Stuck in my
throat.

After murdering
Duncan, he felt his soul in need of God's comfort, yet not surprisingly, it is not
there. Lady Macbeth tries to reason with him. She can sense he is coming unstrung.
Macbeth goes on to say he thought he...

readability="7">

"...heard a voice cry 'Sleep no more! Macbeth
does murder sleep,' / Sleep that knits up the ravelled sleave of care...Still it cried
'Sleep no more...'

Macbeth
believes that because he murdered Duncan in his sleep,
Macbeth will no longer be able to enjoy the rest and rejuvenation
of sleep anymore.

Lady Macbeth shows that "manly" nature
she had prayed for earlier, and disdainfully dresses him down (insults him) for his
unmanly behavior, including bringing the daggers with him. He refuses to return to the
scene of the murder to put the murder weapons back because he is so mentally affected by
the night's events; so Lady Macbeth explains that she will do so and tells him to pull
himself together.

When Lady Macbeth returns, she again
insults her husband saying her hands are now bloodly as well, but she is not the coward
he is. ("I shame to wear a heart so white.") She tells him that if they wash up, they
will feel better, but this murder has damaged Macbeth's heart, soul and sense of
personal morality. For him, washing the blood away will not help
him.

When Macbeth hears the knocking at the gates, he
responds, with seeming regret:

readability="5">

Wake Duncan with thy knocking! I would thou
couldst.

In other words, he
wishes the knocking would wake Duncan—it would mean then that Duncan was still
alive.

By the end of this section, Macbeth has become
haunted by what he has done, and seemingly weaker, and Lady Macbeth has shown how
hard-hearted she is, seemingly much stronger than her husband—at least for
now.