Post 2
Help out one of your classmates by showing what will happen if a lower
weekly salary, but a higher commission rate is used. Choose one of your
classmates posts and adjust their figures as follows.
Cut the weekly salary, w, in half.
Add 4% to the
Post 3
Choose one of your classmates posts and calculate a rough estimate of the total cost of
the shed. Make sure you choose a post that does not already have an average
calculated. One way to find the building cost is using the average cost per square f
Module 5 : Seifert Van kampen Theorem & its application
Lecture 22 : Fundamental groups of certain orthogonal groups
Exercises:
1. Show that the sphere
is isomorphic (as a topological group) to
.
2. Show that the center of the group of non-zero quaternion
Module 3 : Fundamnetal groups and it's basic properties
Lecture 7 : Paths, homotopies and the fundamental group
Exercises:
1. Explicitly construct a homotopy between the loop
sphere
and the constant loop based at
on the
. Note that an explicit formula is
Module 2 : General Topology
Lecture 3 : More Preliminaries from general topology
Exercises:
1. Prove that any continuous function
has a fixed point, that is
such that
to say, there exists a point
.
is connected. Is it true that if
2. Prove that the unit i
Module 2 : General Topology
Lecture 2 : Preliminaries from general topology
Exercises:
1. Prove that a topological space is compact if and only if it satisfies the following condition
known as the finite intersection property. For every family
of closed s
Module 7 : Relative homology,exicism and the Jordan Brouwer separation theorem
Lecture 40 : Inductive limits
Exercises:
1. Prove lemma (40.1)
2. Show that the Prfer group (40.3) is the inductive limit of the sequence of multiplicative
cyclic groups
of ord
Module 6 : Basic homology theory
Lecture 37 : Test V
1. Calculate the homology groups of the double torus.
2. Show that any homeomorphism of
3. Show that
4. Regard
given by
is not a retract of
onto itself must preserve the boundary.
. Use the lifting crit
Lecture - XL Inductive limits
We have frequently encountered situations where a certain space X is canonically embedded in a
larger space Y . A familiar example the sequence of orthogonal groups and the canonical inclusions
SO(2, R) SO(3, R) SO(4, R) . .
Lecture - XXXV The Mayer Vietoris sequence and its applications
The proof of Mayer Vietoris sequence is reminiscent of the Seifert Van Kampen theorem. While
the Seifert Van Kampen theorem enables us to relate the fundamental group of a union U V in terms
Lecture - XXXVI Maps of spheres
We are now in a position to prove the general Brouwers xed point theorem as well as a few other
surprising results concerning maps of spheres. As demonstrated in lecture 10, these higher dimensional
analogues were inaccessi
Module 7 : Relative homology,exicism and the Jordan Brouwer separation theorem
Lecture 39 : Excisim Theorem
Theorem 38.2 (Excision):
Let
be a pair and
be a subset of
interior of
such that the closure of
. Then, the homomorphism
induced by inclusion
is con
Lectures - XXIX/XXX The singular chain complex and homology groups
The program of developing a calculus of chains is now formalized in this lecture. We introduce
a new algebraic category of chain complexes and maps between them and prove the fundamental
t
Lecture - XXXII The abelianization of the fundamental group
In this lecture we shall establish a basic result relating the fundamental group 1 (X, x0 ) and the rst
homology group H1 (X). The result is elegant and states that H1 (X) is the abelianization o
Lecture - XXVII (Test IV)
1. Use the Seifert Van Kampen theorem to compute the fundamental group of the double torus.
2. Let K be a compact subset of R3 and regard S 3 as the one point compactication of R3 . Show
that 1 (R3 K) = 1 (S 3 K).
3. If C is the
Lecture - XXXIV Small simplicies
Recall that the Goursat lemma in complex analysis is proved by subdividing a triangle into four
smaller triangles determined by the midpoints of the sides of the given triangle. The integral over
the given triangle is then
Module 2 : General Topology
Lecture 4 : Further preliminaries from general topology
Exercises
1. What happens if we omit the surjectivity hypothesis on the function
the definition of quotient topology on
induced by
?
by collapsing its
2. Show that the spa
Module 2 : General Topology
Lecture 5 : Topological groups
Exercises:
1. Show that in a topological group, the connected component of the identity is a normal
subgroup.
2. Show that the action of the group
on the sphere
given by matrix
multiplication is t
Module 5 : Seifert Van kampen Theorem & its application
Lecture 25 : Adjunction Spaces
Exercises:
1. We have obtained
by attaching
constant map on the boundary of
an adjunction space.
2. Show that if
and
to a singleton with the attaching map as the
. Disc
Module 4 : Theory of Covering Spaces
Lecture 20 : Orbit Spaces
Exercises:
is a finite group acting freely on a Hausdorff space then the action is
1. Suppose that
properly discontinuous and hence deduce that the group action in the example of the
generaliz
Module 4 : Theory of covering spaces
Lecture 19 : Deck transformations
Exercises:
1. Suppose that
and
are topological groups and
projection that is also a group homomorphism then ker
is a covering
Deck
.
2. Determine the deck transformations for the cove
Module 4 : Theory of covering spaces
Lecture 17 : Action of the fundamental group on the fibers
Exercises:
1. Describe a path in
.
2. Let
whose image under the standard map represents the generator of
be the unit circle in the complex plane and
roots of u
Module 4 : Theory of Covering Spaces
Lecture 18 : The lifting criterion
Exercises:
1. For the map
in example (18.3) show that
is the map
given by
.
2. Suppose
is a path connected topological group with unit element
is a covering map. For any choice of
gro
Module 6 : Basic Homology Theory
Lecture 36 : Maps of Spheres
Exercises:
1. Show that if
and
are two reflections (each with respect to a coordinate plane)
then they are conjugate by a homeomorphism. Deduce that both
and
have degree
-1.
misses a point of
2
Module 4 : Theory Covering Spaces
Lecture 15 : Covering Projections
Exercises:
given by
1. Explain why the map
is not a covering projection?
2. Show that the map
given by
is a covering projection for
.
every
3. Suppose
is a covering projection and
is a cl
Module 3 : Fundamental groups & its basic properties
Lecture 10 : Brouwer's theorem and its applications
Exercises:
1. Suppose that a space
has the fixed point property, is it necessary that it be connected?
Does it have to be path-connected?
2. Explain w
Module 3 : Fundamnetal groups and it's basic properties
Lecture 12 & 13 :The fundamental group of the circle.
Exercises:
1. Formulate and prove the Borsuk Ulam theorem for continuous maps from
line.
to the real
2. Use the Borsuk Ulam theorem to prove that
Module 3 : Fundamental groups & its basic properties
Lecture 11 : Homotopies of maps. Deformation retracts
Exercises:
1. Check that the map
constructed in the proof of theorem 11.3 is continuous and is
indeed a homotopy. Work out the proof of theorem 11.5
Module 3 : Fundamental groups & its basic properties
Lecture 9 : Functorial properties of the fundamental group
Exercises:
1. Show that the sphere
from
retracts onto one of its longitudes. If
is the space obtained
by taking its union with a diameter, ther
Module 4 : Theory of covering space
Lecture 16 : Lifting of paths and homotopies
Exercises:
1. Use the general results of this section to give an efficient and transparent proof that
. First show that for any loop based at , the map
given by
is well defin
Abstract
Substance abuse is a core problem facing the young generation in both developed and
developing countries. The trend of new substances getting into the market is escalating with a
constant positive factor. The consequences of substance abuse does
Running head: 4Ps versus value approach
Title
Name of the student
Course title
4Ps versus value approach
4Ps marketing tactics are seen as a traditional way of approach to marketing practice and
are mainly aimed at the product, unlike value approach which
NETWORK DIAGRAM (CHICAGO CORPORATE SITE): DEFENCE IN DEPTH
Network Diagram (Chicago Corporate Site): Defence in Depth
Student Name
University
1
NETWORK DIAGRAM (CHICAGO CORPORATE SITE): DEFENCE IN DEPTH
Network diagram for Chicago Corporate site
Required
Point estimate has been applied in statistics in which it has been used to estimate parameter of a
given statistic population. Interval points are in many times replaced by interval estimate termed
as the confidence intervals. For example, from question n