Binary Integer Programming
The California Manufacturing Co. is
considering expansion by building a new
factory in either Los Angeles or SanFrancisco, or perhaps even in both cities.
It is also considering building at most one
new warehouse but the choice
Climate in the Calculus Classroom
Thomas J. Pfaff
Ithaca College
[email protected]
There is a great need for scientific research on climate instability and the related energy issues,
and the mathematics community can and should be involved. But we also ne
quantity calculus
Algebra with quantities where the symbols of quantities represent products
of numerical values and their units.
G.B. 3
IUPAC Compendium of Chemical Terminology
2nd Edition (1997)
Chapter
2
INVERSE TRIGONOMETRIC
FUNCTIONS
v Mathematics, in general, is fundamentally the science of
self-evident things. FELIX KLEIN v
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f 1, exists if f is one-one
Lecture 7: Relations
1
Relation
Relation between two objects signify some connection between them. For example,
relation of one person being biological parent of another. If we take any two
people at random, say persons X and Y , then either X is a parent
Lecture 5: Functions : Images, Compositions,
Inverses
1
Functions
We have all seen some form of functions in high school. For example, we have
seen polynomial, exponential, logarithmic, trigonometric functions in calculus.
These functions map real numbers
Lecture 9: Principle of Mathematical Induction
1
Properties of Natural Numbers
Most fundamental property of natural numbers is ability to do proof by induction.
The set of natural numbers is denoted by the set N. The set N consists of a distinguished elem
Lecture 8: Equivalence Relations
1
Equivalence Relations
Next interesting relation we will study is equivalence relation.
Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation
on A. The relation is an equivalence relation if it is re
Lecture 6: Functions : Injectivity, Surjectivity, and
Bijectivity
1
Injectivity, Surjectivity, Bijectivity
We are interested in finding out the conditions for a function to have a left inverse,
or right inverse, or both.
Definition 1.1. Let f : A B be a f
Lecture 2: Strategies for Proofs
1
Introduction
Thales of Miletus of sixth century BC is credited with introducing the concepts of
logical proof for abstract propositions. Euclid popularized the axiomatic system
of proofs in his book The elements, written
Lecture 17: Continuous Functions
1
Continuous Functions
Let (X, TX ) and (Y, TY ) be topological spaces.
Definition 1.1 (Continuous Function). A function f : X Y is said to be
continuous if the inverse image of every open subset of Y is open in X. In othe
Lecture 13: Basis for a Topology
1
Basis for a Topology
Lemma 1.1. Let (X, T) be a topological space. Suppose that C is a collection of
open sets of X such that for each open set U of X and each x in U , there is an
element C C such that x C U . Then C is
QAM-I Assignment (2014-15)
The Cutting Stock Problem
A company manufactures electrical transformers. Transformer core contains a number of metal rods of
a specified length, which varies for different models of transformers depending on the specifications.
A
B
C
1 Paper recycling
2
Cost per ton % of pulp lef
3
De-inking
$20
90%
4
Asphalt dispersion
$15
80%
5
6
Box board
Tissue paper
7
Cost per ton
$5
$6
8
Pulp content
15%
20%
9
10
11 Tons of each paper type processed into grades using de-inking
Box board
Ti
Paper Recycling
A paper recycling plant processes box board, tissue paper, newsprint,
and book paper into pulp that can be used to produce three grades of
recycled paper. The prices per ton and the pulp contents of the four
inputs are shown in the Table b
Queuing Models
The single server waiting line system
Undefined and constant service time.
Finite Queue length.
Finite calling population
The multiple server waiting line.
Single Server waiting line System
The queue discipline: In what order the customers
Goal Programming
Multiple Objectives
The organization instead of single goal, decides
on a list of specific, operationally defined goals
that it wishes to achieve.
The list is arranged from the most important
goal to the goal with the lowest priority.
Pr
Result
Two equivalent system of linear equations have the same set of
solutions.
Result
Two equivalent system of linear equations have the same set of
solutions.
Gaussian Elimination Method: Use the following steps to
solve a system of equations Ax = b.
1
DEPARTMENT OF MATHEMATICS, IIT GUWAHATI
MA101: Mathematics I
Date: August 26, 2011
Quiz I (Maximum Marks: 10)
Time: 8 am - 8:50 am
1. Let A be an n n matrix such that the system of equations Ax = 0 has a non-trivial solution. Is it
possible that the syste
DEPARTMENT OF MATHEMATICS, IIT GUWAHATI
MA101: Mathematics I
Mid Semester Exam (Maximum Marks: 30)
Date: September 20, 2011
Time: 2 pm - 4 pm
1. (a) Prove or disprove: If A and B are two matrices of the same size such that the linear system
of equations A
MA 101
END SEM
MATHEMATICS I
01:0004:00PM
nd
IIT GUWAHATI
22
NOV. 2011
Instructions
1. To print means to write legibly in capital letters.
2. Print your tutorial group:
T
3. Put your signature in the space provided:
4. Print your roll number:
5. Print you
DEPARTMENT OF MATHEMATICS, IIT GUWAHATI
MA101: Mathematics I
Date: August 26, 2011
Quiz I (Maximum Marks: 10)
Time: 8 am - 8:50 am
Model Solutions
1. Let A be an n n matrix such that the system of equations Ax = 0 has a non-trivial solution. Is it possibl
DEPARTMENT OF MATHEMATICS, IIT Guwahati
MA101: Mathematics I, July - November 2011
Errata
Practice Problem Set
4(c).
x = kx0 , y = ky0 .
17. p is a prime number.
44. The first sentence was wrong. We should have A =
"
0 1
1 1
#
.
61. u1 = [1, 2, 2]t , u2 =
MA101:
MATHEMATICS I
[3-1-0-8]
Syllabus and Course Plan
Linear Algebra:
!
"
#
Calculus: Convergence of sequences and series of real numbers; continuity of functions; differentiability,
Rolle's theorem, mean value theorem, Taylor's theorem; power series; R
Lecture 15: The subspace topology, Closed sets
1
The Subspace Topology
Definition 1.1. Let (X, T) be a topological space with topology T. If Y is a
subset of X, the collection
TY = cfw_Y U |U T
is a topology on Y , called the subspace topology. With this
Lecture 11 : Cardinality of Sets
1
Cardinality
We are interested in knowing sizes of sets. Finite sets are usually well behaved.
The difficulty starts in trying to understand infinite sets. Infinite sets have been
notoriously difficult to understand. In f
Lecture 20: Compactness
Parimal Parag
1
Compact spaces
Definition 1.1. A collection A of subsets of a space X is said to cover X, or to
be a covering of X, if the union of the elements of A is equal to X. It is called
an open covering of X if its elements
Lecture 21 : Measurable Spaces
1
Measurable Spaces
Corollary 1.1. Let (X, Fc ) be measurable space and (Y, Fy ) be a topological space
and G be a Borel -algebra on Y . Let g : Y R be a continous function. Then
the map g f : X R is measurable for each meas