This document contains a term end examination for a higher mathematics course. It includes two parts - Part A contains 8 multiple choice or short answer questions, and Part B contains 3 longer answer questions requiring proofs or calculations. Some of the mathematical topics covered include graphs, lattices, modular arithmetic, logic, and probability. The exam is worth a total of 100 marks and students must answer 5 questions in Part A and 3 in Part B.
This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
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This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
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In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
Much has been written in the business literature about managing the waiting experience. Federal Express has noted that “waiting is frustrating, demoralizing, agonizing, aggravating, annoying, time consuming, and incredibly expensive.” We intuitively know this from our own experience as well as from our patients. In this #ACEP13 presentation, Dr. Jensen gives practical tips to improve your patients' ED experience.
Customers Waiting in Lines - Service Operations - Yolanda WilliamsYolanda Williams
The Service Operations of Waiting Lines attempts to remind us that customer needs should be met while they are waiting to be serviced. Also, there is a tremendous reduction in profit by requiring a customer to wait too long. When is too long? That is in the mind of the customer.
Waiting Line Management Problem Solution, Writer Jacobs (1-15)Imran Hossain
This problem solution has been prepared by Abu Zafor, Abdus Salam and Imran Hossain of Islamic University, Kushtia of Management Department, Session: 2010-2011.
Divide-and-Conquer & Dynamic Programming
Divide-and-Conquer: Divide a problem to independent subproblems, find the solutions of the subproblems, and then merge the solutions of the subproblems to the solution of the original problem.
Dynamic Programming: Solve the subproblems (they may overlap with each other) and save their solutions in a table, and then use the table to solve the original problem.
Example 1: Compute Fibonacci number f(0)=0, f(1)=1, f(n)=f(n-1)+f(n-2) Using Divide-and-Conquer:
F(n) = F(n-1) + F(n-2)
F(n-2) + F(n-3) + F(n-3) + F(n-4)
F(n-3)+F(n-4) + F(n-4)+F(n-5) + F(n-4)+F(n-5) + F(n-5) + F(n-6)
…………………….
Computing time: T(n) = T(n-1) + T(n-2), T(1) = 0 T(n)=O(2 ) Using Dynamic Programmin: Computing time=O(n)
n
Chapter 8 Dynamic Programming (Planning)
F(0)
F(1)
F(2)
F(3)
F(4)
……
F(n)
Example 2
The matrix-train mutiplication problem
*
Structure of an optimal parenthesization
Matrix Size
A1 30×35
A2 35×15
A3 15×5
A4 5×10
A5 10×20
A6 20×25
Input
6
5
3
2
4
1
3
3
3
3
3
3
3
3
5
1
2
3
4
5
i
j
1 2 3 4 5
s[i,j]
6
5
3
2
4
i
j
m[i,j]
15125
10500
5375
3500
5000
0
11875
7125
2500
1000
0
9375
4375
750
0
7875
2625
0
15750
0
0
1 2 3 4 5 6
1
i-1
k
j
Matrix-Chain-Order(p)
1 n := length[p] -1;
2 for i = 1 to n
3 do m[i,i] := 0;
4 for l =2 to n
5 do {for i=1 to n-l +1
6 do { j := i+ l-1;
7 m[i,j] := ;
8 for k = i to j-1
9 do {q := m[i,k]+m[k+1,j] +p p p ;
10 if q < m[i,j]
11 then {m[i,j] :=q; s[i,j] := k}; }; };
13 return m, s;
Input of algorithm: p , p , … , p (The size of A = p *p )
Computing time O(n )
8
0
1
n
i
i+1
i
3
Example 3
Longest common subsequence (LCS)
A problem from Bioinformatics: the DNA of one organism may be
S1 = ACCGGTCGAGTGCGCGGAAGCCGGCCGAAA, while the DNA of another organism may be S2 = GTCGTTCTTAATGCCGTTGCTCTGTAAA. One goal of comparing two strands of DNA is to determine how “similar” the two strands are, as some measure of how closely related the two organisms are.
Problem Formulization
Given a sequence X = ( ), another sequence Z = ( ) is a subsequence of X if there exists a strictly increasing sequence ( ) of indices of X such that for all j = 1, 2, …k, we have .
Theorem
Let X = ( ) and Y = ( ...
Digital Signals and System (April – 2017) [75:25 Pattern | Question Paper]Mumbai B.Sc.IT Study
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1. School of Science
Term End Examination- Nov 2009
Subject: Higher Mathematics (CSE 501)
Common to M.Tech-ITNet &CSE
Time : 3 Hrs Max.Marks: 100
PART – A
Answer any 5 of the following Questions 5 x 8 = 40 Marks
1. Obtain the principal disjunctive normal form of ( ) ( )( )P P Q Q P⎯⎯→ → ∧ ¬ ¬ ∨ ¬ .
2. Find and draw all fundamental cycles of the given graph with respect to the spanning
tree T, ( ) { }1 2 3 12 7 6 5e ,e ,e ,e ,e ,e ,e=E T
3. Which of the following posets forms a lattice? Justify briefly your answer
(i) (ii) (iii) (iv)
4. a). Let G be a given simple connected graph on ‘n’ vertices such that each edge
of G lies on atmost one cycle. If G has ‘r’ cycles, find the number of edges of G.
1e
2e
7e
5e
4e 6e8e
9e 10e
12e 11e
3e
a
b c
d
e
k
l
m
n o
f
g
h
I
j
kk
l
o
n
m
2. b).If G represents a graph, whose incidence matrix is given by
I(G) =
0 1 z 1 y 0 1
1 w y 0 0 0 0
0 x 0 0 0 0 0
x 0 z 1 0 y x z
y z x y 0 w w w
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
+⎢ ⎥
⎢ ⎥+⎣ ⎦
Compute the values of x, y, z, w and the adjacency matrix of G.
5. Find the least positive integer x such that x 5(mod7)≡ and x 7(mod11)≡ and
x 3(mod13)≡
6. Show that R (P Q)∧ ∨ is a valid conclusion from the premises P Q∨ , Q R→ ,
P M→ , M¬
7. a) Let G be a simple graph with n vertices and q edges, such that each vertex of G
is of degree either 3 or 4 . Find the number of vertices of degree 4 in G.
b) Let G be a simple connected graph with all its vertices are of degree 3 or 4.
For all edges e {u,v} E(G) d(u) d(v) 7= ∈ ⇔ + = . Does this graph contains any
cycle of ODD length? Justify your answer.
Also prove that
u X v Y
d(u) d(v)
∈ ∈
=∑ ∑ ,
where X = {u V(G) | d(u) 3}∈ = , Y {v V(G) | d(v) 4}= ∈ =
8. At what average rate must a clerk in a supermarket work in order to ensure a probability
of 0.90 that the customer will not wait longer than minutes? It is assumed that there is
only one counter at which customers arrive in a Poisson fashion at an average rate of 15
per hour and that the length of the service by the clerk has an exponential distribution.
3. PART – B
Answer any 3 of the following Questions 3 x 20 = 60 Marks
9. Prove that for any prime p , all the coefficients of the polynomial
p 1
f(x) (x 1)(x 2) (x p 1) x 1−
= − − ⋅⋅⋅ − + − + are divisible by p.
10. i. Let L be a Lattice. Prove that the following statements are true.
1. ∨ ∧a a= a ,a a= a
2. ∨ ∨ ∧ ∧a b= b a ,a b= b a
3. ( ) ( ) ( ) ( )∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧a b c = a b c ,a b c = a b c
4. ( ) ( )∨ ∧ ∧ ∨a a b = a ,a a b = a
ii. Compute ( ) ( ) ( ) ( )a c e f a b c e∨ ∧ ∨ ∨ ∨ ∧ ∨⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
11. i. If T1 and T2 are given two spanning trees of a given connected graph G, then prove
the following :
( ) ( )E( ) E( ) E( ) E( ) E( ) E( )1 2 1 2G T T T G T=− −∩ ∩
ii. Find the Minimum and Maximum Spanning Trees of the following connected
graph.
12. a. Show that ( )x M(x)∃ follows logically from the premises
( )( )x H(x) M(x)⎯⎯→ and ( )x H(x)∃
0
a b
c
e f
I
0v
1 2
9
2
9
3
61
5
6
2
1
8
2
9 1
3
4 1
4
7
7
0u
4. b. Prove that the total number of vertices of degree 2 of a given tree ‘T’
on ‘q’ edges ( 2)≥q is equal to [ ]
v V(T)
(v) 3
( 1) (v) 1
∈
≥
− − −∑q
d
d .
Also prove that the expression
j
j
j
w V(T)
) 3
2 d(w ) 2
d(w
∈
≥
⎡ ⎤+ −⎣ ⎦∑ represents the total number of
pendent vertices of T
13. Customers arrive at a one-man barber shop according to a Poisson
process with a mean inter arrival time of 12 min. Customers spend an
average of 10 min in the barber’s chair.
(a). What is the expected number of customers in the barber shop and in the
queue
(b). Calculate the percentage of time an arrival can walk straight into the
barber’s chair without having to wait.
(c). How much time can a customer expect to spend in the barber’s shop?
(d). Management will provide another chair and hire another barber , when a
Customer’s waiting time in the shop exceeds 1.25 h. How much must
the average rate of arrivals increase to warrant a second barber?
(e). What is the average time customers spend in the queue?
(f). What is the probability that the waiting time in the system is greater than
30 min
(g) Calculate the percentage of customers who have to wait prior to getting
into the barber’s chair.
(h) What is the probability that more than 3 customers are in the system?