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Recent questions tagged discrete-mathematics
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91
Kenneth Rosen Edition 7 Exercise 8.3 Question 13 (Page No. 535)
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
Lakshman Patel RJIT
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May 10, 2020
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Lakshman Patel RJIT
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Kenneth Rosen Edition 7 Exercise 8.3 Question 12 (Page No. 535)
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
Lakshman Patel RJIT
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Combinatory
May 10, 2020
by
Lakshman Patel RJIT
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.3 Question 11 (Page No. 535)
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
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Kenneth Rosen Edition 7 Exercise 8.3 Question 10 (Page No. 535)
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
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Lakshman Patel RJIT
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Kenneth Rosen Edition 7 Exercise 8.3 Question 9 (Page No. 535)
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
224
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96
Kenneth Rosen Edition 7 Exercise 8.3 Question 8 (Page No. 535)
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
311
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Kenneth Rosen Edition 7 Exercise 8.3 Question 7 (Page No. 535)
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
235
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98
Kenneth Rosen Edition 7 Exercise 8.3 Question 6 (Page No. 535)
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$
Lakshman Patel RJIT
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Combinatory
May 10, 2020
by
Lakshman Patel RJIT
207
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99
Kenneth Rosen Edition 7 Exercise 8.3 Question 5 (Page No. 535)
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$-bit integers using the fast multiplication algorithm.
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
162
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kenneth-rosen
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100
Kenneth Rosen Edition 7 Exercise 8.3 Question 4 (Page No. 535)
Express the fast multiplication algorithm in pseudocode.
Lakshman Patel RJIT
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Combinatory
May 10, 2020
by
Lakshman Patel RJIT
204
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kenneth-rosen
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101
Kenneth Rosen Edition 7 Exercise 8.3 Question 3 (Page No. 535)
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.
Lakshman Patel RJIT
asked
in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
249
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102
Kenneth Rosen Edition 7 Exercise 8.3 Question 2 (Page No. 535)
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?
Lakshman Patel RJIT
asked
in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
234
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kenneth-rosen
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103
Kenneth Rosen Edition 7 Exercise 8.3 Question 1 (Page No. 535)
How many comparisons are needed for a binary search in a set of $64$ elements?
Lakshman Patel RJIT
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in
Combinatory
May 10, 2020
by
Lakshman Patel RJIT
268
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Kenneth Rosen Edition 7 Exercise 8.2 Question 52 (Page No. 527)
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... solution of the form $n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
269
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kenneth-rosen
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105
Kenneth Rosen Edition 7 Exercise 8.2 Question 51 (Page No. 527)
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$ has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
217
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kenneth-rosen
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106
Kenneth Rosen Edition 7 Exercise 8.2 Question 53 (Page No. 527)
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
242
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kenneth-rosen
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107
Kenneth Rosen Edition 7 Exercise 8.2 Question 50 (Page No. 527)
It can be shown that Cn, the average number of comparisons made by the quick sort algorithm (described in preamble to question $50$ in exercise $5.4),$ when sorting $n$ ... $48$ to solve the recurrence relation in part $(A)$ to find an explicit formula for $C_{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
146
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108
Kenneth Rosen Edition 7 Exercise 8.2 Question 49 (Page No. 527)
Use question $48$ to solve the recurrence relation $(n + 1)a_{n} = (n + 3)a_{n-1} + n, \:\text{for}\: n \geq 1, \:\text{with}\: a_{0} = 1$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
114
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.2 Question 48 (Page No. 526)
Some linear recurrence relations that do not have constant coefficients can be systematically solved. This is the case for recurrence relations of the form $f (n)a_{n} = g(n)a_{n-1} + h(n).$ Exercises $48-50$ ...
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
235
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Kenneth Rosen Edition 7 Exercise 8.2 Question 47 (Page No. 526)
A new employee at an exciting new software company starts with a salary of $\$50,000$ and is promised that at the end of each year her salary will be double her salary of the previous year, with an extra increment of $\ ... year of employment. Solve this recurrence relation to find her salary for her $n^{\text{th}}$ year of employment.
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
280
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111
Kenneth Rosen Edition 7 Exercise 8.2 Question 46 (Page No. 526)
Suppose that there are two goats on an island initially.The number of goats on the island doubles every year by natural reproduction, and some goats are either added or removed each year. Construct a recurrence relation for the number of goats on ... assuming that n goats are removed during the $n^{\text{th}}$ year for each $n \geq 3.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
279
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.2 Question 45 (Page No. 526)
Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of $1$ month and six new pairs of rabbits at the age of $2$ months and every month afterward. ... $n$ months after one pair is left on the island.
Lakshman Patel RJIT
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in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
182
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.2 Question 44 (Page No. 526)
(Linear algebra required ) Let $A_{n}$ be the $n \times n$ matrix with $2s$ on its main diagonal, $1s$ in all positions next to a diagonal element, and $0s$ everywhere else. Find a recurrence relation for $d_{n},$ the determinant of $A_{n}.$ Solve this recurrence relation to find a formula for $d_{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
137
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kenneth-rosen
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114
Kenneth Rosen Edition 7 Exercise 8.2 Question 43 (Page No. 526)
Express the solution of the linear nonhomogenous recurrence relation $a_{n} = a_{n-1} + a_{n-2} + 1\:\text{for}\: n \geq 2 \:\text{where}\: a_{0} = 0\:\text{and}\: a_{1} = 1$ in terms of the Fibonacci numbers. [Hint: Let $b_{n} = a_{n + 1}$ and apply question $42$ to the sequence $b_{n}.]$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
99
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Kenneth Rosen Edition 7 Exercise 8.2 Question 42 (Page No. 526)
Show that if $a_{n} = a_{n-1} + a_{n-2}, a_{0} = s\:\text{and}\: a_{1} = t,$ where $s$ and $t$ are constants, then $a_{n} = sf_{n-1} + tf_{n}$ for all positive integers $n.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
83
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Kenneth Rosen Edition 7 Exercise 8.2 Question 41 (Page No. 526)
Use the formula found in Example $4$ for $f_{n},$ the $n^{\text{th}}$ Fibonacci number, to show that fn is the integer closest to $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$ Determine for which $n\: f_{n}$ is greater ... for which $n\: f_{n}$ is less than $\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$
Lakshman Patel RJIT
asked
in
Combinatory
May 6, 2020
by
Lakshman Patel RJIT
114
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.2 Question 40 (Page No. 526)
Solve the simultaneous recurrence relations $a_{n} = 3a_{n-1} + 2b_{n-1}$ $b_{n} = a_{n-1} + 2b_{n-1}$ with $a_{0} = 1 \: \text{and}\: b_{0} = 2.$
Lakshman Patel RJIT
asked
in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
584
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Kenneth Rosen Edition 7 Exercise 8.2 Question 39 (Page No. 526)
a) Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = a_{n-4}.$ [Note: These include complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1, a_{1} = 0, a_{2} = -1,\: \text{and}\: a_{3} = 1.$
Lakshman Patel RJIT
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in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
124
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Kenneth Rosen Edition 7 Exercise 8.2 Question 38 (Page No. 526)
Find the characteristic roots of the linear homogeneous recurrence relation $a_{n} = 2a_{n-1} - 2a_{n-2}.$ [Note: These are complex numbers.] Find the solution of the recurrence relation in part $(A)$ with $a_{0} = 1\:\text{and}\: a_{1} = 2.$
Lakshman Patel RJIT
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in
Combinatory
May 5, 2020
by
Lakshman Patel RJIT
458
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 8.2 Question 37 (Page No. 526)
Let an be the sum of the first $n$ triangular numbers, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} t_{k},\:\text{where}\: t_{k} = k(k + 1)/2.$ Show that $\{an\}$ satisfies the linear nonhomogeneous ... and the initial condition $a_{1} = 1.$ Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
Lakshman Patel RJIT
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Combinatory
May 5, 2020
by
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147
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