OUTLINE
Network Flows
I
Preflows
F
F
I
General Scheme of a Preflow Algorithm
Ahuja & Orlin Algorithm
Combinatorial Applications
F
F
F
F
Bipartite Matchings
Digraphic Degree Sequences
Edge Connectivity
Vertex Connectivity
2 / 34
Network Flows
Maximum Flow
1. Let functions f, g, h from V = cfw_1, 2, 3, 4 into V be defined by:
f (n) = 6 n,
g(n) = 3,
h = cfw_(1, 2), (2, 3), (3, 4), (4, 1)
Decide which functions are: one-to-one, onto, or invertible.
SOLUTION:
(i)
We have
f : V V V =cfw_1,2,3,4
f ( n )=6n
Subs
SOLUTION: (a)
2
2
ex /2 dx
We have
1
Using midpoint rule
b
f ( x ) dx= x
a
Where
Taking
x=
x +x
x +x
x +x
x
+f (
+f (
+f (
(
)
)
)
[ 2 2 2
f
0
1
1
2
2
3
n2
+ x n1
x +n
+ f n1
2
2
) (
ba
n
a=1,b=2, n=6, x =
21 1
=
6
6
Divide the interval [1,2] into 6 su
Homework 10
due Friday Dec 2, 10pm (or Friday in class)
Review of determinants and their properties
If A is a square n X n matrix, it represents a linear function R" i) R (with the same
R as input and as output). The determinant of A is dened as the facto
Worksheet 11 Solutions, Math 53
Line Integrals
Wednesday, November 7, 2012
1. If C is a smooth curve given by a vector function r(t), a t b, and v is a constant vector, show that
Z
v dr = v [r(b) r(a)]
C
Solution
Suppose that v = hx0 , y0 , z0 i. Then f (
Undergraduate Research Opportunity Programme in Science
(UROPS)
Jordan Canonical Forms of Linear Operators
Submitted by
Teo Koon Soon
Supervised by
Dr. Victor Tan
Department of Mathematics
National University of Singapore
Academic Year 2001/2002 Semester
SOLUTION: We parametrize C by
r ( t ) =( 1t )( 0,4,7 )+ t ( 4,2,2 )= ( 4 t , 44 t ,75 t )
r ( t ) =( 4 t , 44 t , 75 t )
x=4 t , y=42t , z=75 t ,0 t 1
ds=
(
dx 2 dy 2 dz 2
+
+
dt
dt
dt
dt
)( )( )
ds= ( 4 ) + (2 ) + (5 ) dt= 45 dt
2
2
2
1
x
3
x
3
z ds= ( 4
QUESTION1:
SOLUTION: (a)
We have
2
y=16 t +0.05 t + 168
Diffrentiating with respect to t
dy
=32 t+ 0.5
dt
When the top of the ladder will reach the ground then velocity will be zero
Therefore,
dy
=0
dt
32 t+ 0.5=0
t=
1
se c
64
SOLUTION: (b)
FIRST Method
B
7
Greens Functions and Nonhomogeneous
Problems
The young theoretical physicists of a generation or two earlier subscribed to the
belief that: If you havent done something important by age 30, you never will.
Obviously, they were unfamiliar with the histor
NOTES ON THE EXISTENCE AND UNIQUENESS THEOREM
FOR FIRST ORDER DIFFERENTIAL EQUATIONS
I. Statement of the theorem.
We consider the initial value problem
cfw_ y(w) = F(w,y($)
y($o) = yo.
(1.1)
Here we assume that F is a function of the two variables (x, y),
Chapter 5
Boundary Value Problems
A boundary value problem for a given differential equation consists of finding a solution of the
given differential equation subject to a given set of boundary conditions. A boundary condition
is a prescription some combi
S. Ghorai 1
Lecture V
Picards existence and uniquness theorem, Picards iteration
1 Existence and uniqueness theorem
Here we concentrate on the solution of the rst order IVP
1/ = J00, 1/); 21(900) = 3/0 (1)
We are interested in the following questions:
1.
Existence and Uniqueness Theorems for FirstOrder ODEs
The general rstorder ODE is
y = F(x, y), 31(900) = y0- (*)
We are interested in the following questions:
(i) Under what conditions can we be sure that a solution
to (*) exists?
(ii) Under what conditio
Engineering Mechanics
Prof. Manoj Harbola
Indian Institute of Technology, Kanpur
Module - 05
Lecture - 01
Motion of Particles Planar Polar Co-ordinater
So, for we have been dealing with statics, this is the first lecture in dynamics.
(Refer Slide Time: 00
Indian Institute of Technology Bhubaneswar
Details regarding the Registration of B. Tech Fresher Candidates
Dear Candidate,
Congratulations on qualifying in JEE (Advanced) 2016 and choosing IIT Bhubaneswar for your
studies. You have been PROVISIONALLY s
Engineering Mechanics
Prof. Manoj Harbola
Indian Institute of Technology, Kanpur
Module - 03
Lecture - 03
Properties of Surfaces - III
In the previous lecture, we have been talking about the first moment of a plane area and a
centroid. Continuing on that
Assignment 2
1. Find out expression for corrosion rate from Faradays laws of electrochemistry for
(a) Uniform corrosion and (b) pitting corrosion.
2. Find out multiplication factor for the conversion of corrosion rate from
(a) mdd to mpy, (b) mpy to mmy-1
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HACK
Speed Up Boot and Shutdown Times
H A C K
#3
Speed Up Boot and Shutdown Times
Hack #3
Shorten the time it takes
Jom J Kandathil | ME13B111
Indian Institute of Technology Madras
EDUCATION
Program
Dual degree in Mechanical
Institution
%/CGPA
Indian Institute of Technology Madras, Chennai
8.66/10
2018
Kendriya Vidyalaya Ernakulam
Kendriya Vidyalaya Ernakulam
94.2
9.5/
Assignment 5
1. What is oxidation?
2. Oxidation is an example of dry corrosion: Justify.
3. What is Pilling-Bedworth ratio? What are its significance and limitations in predicting
oxidation resistance of a metal?
4. How would partial pressure of O2 relate
Softwares required :
1) Putty
2) Bit comet
3) Chemical department user ID
Putty settings :
1)session-host name port
127.0.0.1 3535
protocol-ssh
tunnels-dynamic 5656 press add
Go back to sessions and save as in
2) Restart putty.
3)session-host name port
10
Hussain. H . Khat
Third Year Undergraduate Student
Mechanical Engineering
Indian Institute Of Technology Madras
Chennai 36
India.
Email: hussain.khat@gmail.com
Career Objective:
Be a loyal and hardworking employee.
Utilize my existing knowledge and learn