There are many issues revolving around investor protection in the world. To start with countries with
common law tradition such as Canada, UK and US have highly protective investors policies compared
1.
a.
b.
c.
d.
Chapter 8
Global Warming In April 2011, a Yale/George Mason poll of 1010 US adults found that 40% of the
people responding believed that scientists disagreed about whether global warmin
For the Unit VIII assignment, please refer to Section 5.4 of the text.
Monica works at a regional weather office on the Atlantic coast. She notes (from the office records) that
hurricanes have made la
You work for a HR consulting company and an organization (the same company you
have been writing about during this course) has hired your firm to conduct an HRM
analysis and make recommendations to be
ENGR 112
HOMEWORK ASSIGNMENT #6
1. Evaluate the following SCILAB expressions yourself (Dont use SCILAB to
help! This is working on the concept of precedence). You can do this all in
SCILAB, but for th
Executive Summary
Recently, the Genesis Energy is going to make a contract with Sensible Energy to undertake the
project of operating expansion. But before going along with this project, study of proj
CMN 251: Organizational Communication
Final Case Development & Analysis Paper
For this assignment, you are required to create and analyze an original case.
Working with 1-2 of your peers, you will nee
Physical Science Final Exam
Name:
Student Number:
Type your answers in the blank for each question. Do NOT use the internet - use ONLY the textbook to
create solutions in your own words.
1. A car that
Lecture 7
Morphisms of projective algebraic
varieties
Following the denition of a morphism of a ne algebraic varieties we can dene a
morphism f : X ! Y of two projective algebraic varieties as a set o
Lecture 6
Bzout theorem and a group law
e
on a plane cubic curve
We begin with an example. Consider two concentric circles:
2
2
C : Z1 + Z2 = 1 ,
2
2
C 0 : Z1 + Z2 = 4 .
Obviously, they have no common
48LECTURE 6. BEZOUT THEOREM AND A GROUP LAW ON A PLANE CUBIC CURVE
the line T0 + T1 + T2 = 0 contains a given point (resp. two distinct points)
is a two-dimensional (resp. one-dimensional) linear sub
50LECTURE 6. BEZOUT THEOREM AND A GROUP LAW ON A PLANE CUBIC CURVE
Then we extend the denition to all polynomials by linearity over k requiring that
@ (aP + bQ)
@P
@Q
=a
+b
@ Zj
@ Zj
@ Zj
for all a,
55
T0 T1 T2 = 0 which contains the points (1, 0, ), (1, , 0). The set x1 , . . . , x8
is the needed conguration. One easily checks that the nine points x1 , . . . , x9
are the inection points of the c
53
x
y
xy
o
x
y
Figure 6.1:
line to X at e. If y = x, take for L1 the tangent line at y . We claim that this
construction denes the group law on X (K ).
Clearly
y x = x y,
i.e., the binary law is comm
44
LECTURE 5. PROJECTIVE ALGEBRAIC VARIETIES
In future we will always assume that a projective variety X is given by a
system of equations S such that the ideal (S ) is saturated. Then I = (S ) is
den
43
Clearly I sat is a homogeneous ideal in k [T ] containing the ideal I (Check it !) .
Proposition 5.14. Two homogeneous systems S and S 0 dene the same projective variety if and only if (S )sat = (S
33
the eld of fractions: it is the set of equivalence classes of pairs (m, s) 2 M
S with the equivalence relation: (m, s) (m0 , s0 ) () 9s00 2 S such that
s00 (s0 m sm0 ) = 0. The equivalence class o
Lecture 5
Projective algebraic varieties
Let A be a commutative ring and An+1 (n
0) be the Cartesian product
equipped with the natural structure of a free A-module of rank n + 1. A free
submodule M of
35
Proof. Let Matn (A) be the ring of n n matrices with coe cients in a commutative ring A. For any ideal I in A we have a natural surjective homomorphism
of rings Matn (A) ! Matn (A/I ), X 7! X , whi
37
A is a eld this is a familiar duality between lines in a vector space V and
hyperplanes in the dual vector space V .
A set cfw_fi i2I of elements from A is called a covering set if it generates
the
41
Example 5.12. Let X be given by aT0 + bT1 + cT2 = 0, a projective subvariety
of the projective plane P2 . It is equal to the projective closure of the line L A2
k
k
given by the equation bZ1 + cZ2
39
Denition 5.4. A polynomial F (T0 , . . . , Tn ) 2 k [T0 , . . . , Tn ] is called homogeneous of degree d if
X
X
i
i
F (T0 , . . . , T n ) =
ai0 0,.,in 0 T00 Tnn =
ai T i
i0 ,.,in
i
with |i| = d for
23
Proof. Let us prove the rst part. If V is irreducible, then the assertion is obvious.
Otherwise, V = V1 [ V2 , where Vi are proper closed subsets of V . If both of
them are irreducible, the asserti
15
Proof. Let f : X ! Y be a morphism. It denes a homomorphism
fO(X ) : Homk (O(X ), O(X ) ! Homk (O(Y ), O(X ).
The image of the identity homomorphism idO(X ) is a homomorphism : O(Y ) !
O(X ). Let u
17
algebras: O(Y ) = k [T1 ] ! O(X ). Every such homomorphism is determined by
its value at T1 , i.e. by an element of O(X ). This gives us one more interpretation
of the elements of the coordinate al
19
X ! Y of algebraic varieties denes a map fK : X (K ) = V ! Y (K ) =
W of the algebraic sets. So it is natural to take for the denition of regular
maps of algebraic sets the maps arising in this way
Lecture 4
Irreducible algebraic sets and
rational functions
We know that two a ne algebraic k -sets V and V 0 are isomorphic if and only if
their coordinate algebras O(V ) and O(V 0 ) are isomorphic.
Lecture 3
Morphisms of a ne algebraic
varieties
In Lecture 1 we dened two systems of algebraic equations to be equivalent if
they have the same sets of solutions. This is very familiar from the theory
11
Next we shall show that the set of algebraic k -subsets in K n can be used
to dene a unique topology in K n for which these sets are closed subsets. This
follows from the following:
Proposition 2.7
Lecture 1
Systems of algebraic equations
The main objects of study in algebraic geometry are systems of algebraic equations and their sets of solutions. Let k be a eld and k [T1 , . . . , Tn ] = k [T