Discuss the relation between Hobbes view of human nature and his account of the origin, character and functions of the State.

Hobbes's belief that human nature is inherently selfish
and prone to brutality informs his vision of what the state should be like.  Because we
are naturally "bad," Hobbes believes that we need to totally subjugate ourselves to the
state so that it may protect us from our evil natures (and those of our fellow human
beings).

Hobbes argues that life in the state of nature is
chaotic.  It is the war of each against all.  If we are to avoid this fate, we have to
voluntarily submit to an absolutist state.  We have to allow the state to have complete
control over us so that it can ensure that we will not revert to the state of nature. 
As the link below tells us,

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In Hobbes’s view, stability could be achieved
only where sovereignty was undivided and absolute. Otherwise civil authority would come
undone, and the brutishness of the state of nature would reassert
itself.

What are the elements of Petrarchanism in Romeo and Juliet?

Francesco Petrarch (1304-1374), an early Renaissance poet,
wrote more than 300 sonnets addressed to Laura, a lady forever unobtainable. Thus,
he established the model of love poetry that prevailed for the next two centuries,
imitated by many writers, Shakespeare included. Along with its prescribed form, the
Petrarchan sonnet had the following conventions:

The lady
to whom the poem is addressed is forever beyond reach - she is on a pedestal; the poet's
love for her is unrequited; the poet is sick to death with love; the lady is described
with encomiums almost insincere; and the poet's love is expressed in religious
terms. 

However, Shakespeare was not a slave to the
traditions of the Petrachan form. He breaks with that tradition by burying one Petrachan
sonnet and the first four lines of a second within the dramatic action of Romeo's first
spoken encounter with Juliet in Act 1, Scene 5, lines 93 to
110.   

Romeo begins his Juliet wooing with four lines of
conventional Petrarchan language; and there is religious terminology throughout their
exchange. But in the second quatrain (lines 97-100), spoken by the poet's lady, it is
very clear that Juliet is not aloof - no pedestal for her. In their verbal foreplay
before the first kiss, Juliet matches her wooer word for word, line for line. She and
Romeo are very well aware of the conventions and operate within them with dexterity, but
perhaps galvanized by the headiness of forbidden love, they strive to break them - just
as Shakespeare both conformed to and transcended the Petrachan form. As they complete
their trysting first sonnet and begin another (at line 107), it is clear that Juliet is
a new kind of sonnet-heroine, audacious and eager for love, not distant and
unattainable, like a malicious diva. Cast not as a romantic set-piece, but as an
integral part of the dramatic action, this sonnet of Act 1, Scene 5 leads directly into
the first quatrain of the second (lines 107-110), broken, ironically, by the arrival of
the Nurse. Both sonnets, therefore, unmistakably foreshadow the beginning of true
love and its all-too-soon ending by tragic death.  

Given u = (1-m)*i + 1+m)*j and v=(1+2m)*i+4j determine the vectors u and v whether u is perpendicular to v.

By definition, two vectors are perpendicular if the value
of their dot product is zero.

u*v = 0
(1)

[(1-m)*i+(1+m)*j]*[(1+2m)*i+4*j] = (1-m)*(1+2m) +
4*(1+m) (2)

We'll equate (1) and
(2):

(1-m)*(1+2m) + 4*(1+m) =
0

We'll remove the brackets:

1
+ 2m - m - 2m^2 + 4 + 4m = 0

We'll combine like
terms:

-2m^2 + 5m + 5 =
0

We'll multiply by -1:

2m^2 -
5m - 5 = 0

We'll apply quadratic
formula:

m1 =
[5+sqrt(25+40)]/4

m1 =
(5+sqrt65)/4

m2 =
(5-sqrt65)/4

The values of the parameter "m",
for the vector u to be perpendicular to the vector v, are: {(5-sqrt65)/4 ;
(5+sqrt65)/4}.

In Guns, Germs, and Steel, why does the author talk of "east-west axis" and "north-south axis?"

Diamond talks about north-south and east-west axes because
they have a great impact on how much diffusion of plants there can be in any given area
of the world.

If an area of the world is like the Americas,
its long axis runs north to south.  That means that it is longer north to south than it
is east to west.  Eurasia is the opposite because it is longer east to west than it is
north to south.  It is easier for plants to diffuse along an east-west axis.  This is
because climate does not change east to west (outside of factors like elevation and
distance from an ocean) the way it does from north to
south.

This is important because a society that developed
agriculture could diffuse its domesticated plants along an east-west axis to other
cultures.  That means that civilizations could grow and could exchange new plants with
one another.  In a place with a long north-south axis, this could not
occur.

This is one major reason why Eurasia was so much
better suited to creating major civilizations than places like Africa and the Americas
were.

Find these two regionalist elements in the story "The Outcasts of Poker Flat."Find and explain one example of each of these two regionalist...

Much like some of Mark Twain's narratives, Bret Harte's
"The Outcasts of Poker Flat" is a Western story which employs regionalisms, expressions
that are distinctly Western.  In addition, he wrily makes use of euphemisms to suggest
the hypocrisy of the residents of Poker Flat.  One such
euphemism occurs in the sixth paragraph of Harte's
narrative in which the narrator refers to the "certain objectionable characters" whom
the residents blame for their misfortunes.  These characters are the gambler, Mr.
Oakhurst; a woman suspected of being a witch, Mother Shipton; Uncle Billy, a drunkard
and a "sluice robber," a thief who takes gold from long troughs used for sifting gold;
and a prostitute called Duchess. This group is euphemistically termed,
"the deported wickedness of Poker
Flat."

An example of comic
irony occurs in the eighth paragraph in which the narrator describes the
place to which the outcasts must migrate:

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The road to Sandy Bar--a camp that, not having as
yet experienced the regenerative influences of Poker
Flat....

This phrase means
that Sandy Bar has not yet lost enough money to gamblers and such that it
has "regenerated" its moral code as has Poker Flat. This allusion to the "regenerative
influences" refers to the "emigrants" who are expelled from Poker Flat.  Here Bret Harte
makes fun of the "spasm of virtuous reaction" [par.3] (also
comic irony) which is brought about by the financial losses dealt the residents who
have engaged in activites with the witch, prostitute, or gambler. The loss of money,
etc. has caused the townspeople to angrily seek revenge against these immoral persons by
expulsion from town; that is, the hypocritically sanctimonious expulsion of the people
who have simply taken the money  they have of free choice spent on their own greed or
prurient desires. 

What's a factor of x^3 - x - 6 =0

We will substitute with x=
2.

==> x^3 - x - 6 =
0

==> 8 - 2 -6 = 0

Then
x= 2 is a root for the equation.

==> (x-2) is a
factor.

Now we will divide by (x-2) to find other
factors.

==> (x^3 - x - 6) = (x-2) (x^2 + 2x+
3)

Now sine delta = (4-4*3) = 4-12 = -8 <
0

Then there are no Real roots foe the quadratic
equation.

Then the factors of the equation
are : (x-2) and (x^2 + 2x+3)

If tanx=1/2 what is tan(x+pi/3)

We'll recall the following identity concerning the tangent
of the sum of two angles:

tan(a+b) = (tan a + tan
b)/[1-(tan a)*(tan b)]

Comparing, we'll
get:

tan (x + pi/3) = (tan x + tan pi/3)/[1-(tan x)*(tan
pi/3)]

But tan pi/3 = sqrt3 and tan x = 1/2, therefore,
we'll have;

tan (x + pi/3) = (1/2 +
sqrt3)/[1-(1/2)*(sqrt3)]

tan (x + pi/3) = [(1 +
2sqrt3)/2]/[(2-sqrt3)/2]

tan (x + pi/3) = (1 +
2sqrt3)/(2-sqrt3)

We'll multiply by the conjugate of
denominator:

tan (x + pi/3) = (1 +
2sqrt3)(2+sqrt3)/(2-sqrt3)(2+sqrt3)

The product from
denominator returns the difference of two
squares:

(2-sqrt3)(2+sqrt3) = 4 -
3

tan (x + pi/3) = (1 +
2sqrt3)(2+sqrt3)/(4-3)

tan (x + pi/3) = 2 + sqrt3 + 4sqrt3
+ 6

tan (x + pi/3) = 8 +
5sqrt3

The tangent of the sum x + pi/3 is tan
(x + pi/3) = 8 + 5sqrt3.