... important proof technique in computer science. Generally,
induction is used to prove that some statement holds for all natural values of a variable.
For example, here is a classic formula:
46 Number Theory ... 1) for all n ∈ N.
By the principle of induction, P (n) is true for all n ∈ N, which proves the claim.
This proof would look quite mysterious to anyone not privy to the scratchwork we did
beforehand. ... divisibility hold.
1. If a | b, then a | bc for all c.
2. If a | b and b | c, then a | c.
3. If a | b and a | c, then a | sb + tc for all s and t.
4. For all c = 0, a | b if and only if ca | cb.
Proof....
... original formula for q. Recall that our proof of the formula we had in
Exercise 1.4-5 did not explain why the product of three factorials appeared in the denominator,
it simply proved the formula ... distinct elements. There are n choices for the first number in the list.
For each way of choosing the first element, there are n −1choices for the second. For each choice
of the first two elements, ... members (assuming that the president is not on the advisory
board)?
10. By using the formula for
n
2
is is straightforward to show that
n
n − 1
2
=
n
2
(n − 2).
However this proof just uses...
... 170
Introduction to Programming
3
Computers have a fixed set of instructions that they can perform for us. The specific
instruction set depends upon the make and model of a computer. However, these instructions ... that the computer
always attempts to do precisely what you tell it to do. Say, for example, you tell the computer to
divide ten by zero, it tries to do so and fails at once. If you tell the computer ... instructions that tell the computer
every step to take in the proper sequence in order to solve a problem for a user. A programmer
is one who writes the computer program. When the computer produces a...
... are quite non-trivial to execute on a modern digital computer, but
this algorithm wasn’t designed for a digital computer; it was designed for the Platonic Ideal Classical
Greek Mathematician, ... commercial distribution is expressly forbidden. See http://www.cs.uiuc.edu/~jeffe/teaching/algorithms/ for the most recent revision.
9
Algorithms
Department of Computer Science
University of Illinois ... y
instead of
x ∗ y
for multiplication; write
x mod y
instead of
x
%
y
for remainder; write
x
instead
of
sqrt
(
x
) for square roots; write
a
b
instead of
power
(
a, b
) for exponentiation;...
... telephone)?
Which informations we may consider for the future (e.g.
email, birthday, bankaccount, webpage, ip, image, holographic
picture, etc )?
By means of which information should we sort ... course
Historical development of databases
Databases
Introduction
Michael Emmerich
Leiden Institute for Advanced Computer Science, Leiden
University
January 17, 2012
Michael T. M. Emmerich Databases
Preliminaries
Introductory ... Paradox, Dbase-III (later FoxPro),
System R/R+, IBM-DB2, Watcom SQL, etc.
Simple databases for personal computer arise, such as
Excel/Access
1990ties: The internet emerges and with it web-based
database...
...
In this problem, we will be reading in formatted data and generating a report. One of the
common formats for interchange of formatted data is ’tab delimited’ where each line corresponds
to ...
}
6
MIT OpenCourseWare
http://ocw .mit. edu
6.087 Practical Programming in C
January (IAP) 2010
For information about citing these materials or our Terms of Use, visit: http://ocw .mit. edu/terms
. ... characters.
4
Massachusetts Institute of Technology
Department of Electrical Engineering and ComputerScience
6.087: Practical Programming in C
IAP 2010
Problem Set 3
Control flow. Functions....
... degree-of-freedom pose space for a 3D object and also does not account for any non-
rigid deformations. Therefore, we use broad bin sizes of 30 degrees for orientation, a factor
of 2 for scale, and 0.25 ... operations are performed
on image data that has been transformed relative to the assigned orientation, scale, and
location for each feature, thereby providing invariance to these transformations.
4. ... while
allowing for shifts in their position results in much better classification under 3D rotation. For
example, recognition accuracy for 3D objects rotated in depth by 20 degrees increased from
35% for...
... values in common to both sets.
17
49
COMPUTER SCIENCE TEST
PRACTICE BOOK
Score Conversions and Percents Below*
For GRE ComputerScience Test, Form GR0329 Only
TOTAL SCORE
Raw Score Scaled ... to
compare your performance with the performance of
others who took this test. Both the worksheet on page
48 and the table on page 49 use performance data from
GRE ComputerScience Test examinees.
The ... Scores* Needed to
Earn Selected Scaled Scores on
Three ComputerScience Test Editions
That Differ in Difficulty
Raw Scores
Scaled Score Form A Form B Form C
Number of Questions Used to Compute Raw Score
*Raw...
... book is an introduction to computer science. Computerscience is the study of
computer hardware, algorithms, and data structures and how they fit together to provide
information systems. Each of ... sum-
maries, and animations. Computers are therefore information-
processing machines, and the computer programs are
information-processing systems.
1.1 A BRIEF HISTORY OF COMPUTING
Computers as we know ... widespread use was FORTRAN (short for formula translation system).
Released in 1954 by IBM, FORTRAN was designed for scientific (mathematical) pro-
gramming and allowed mathematical formulas to be...
...
Faculty of ComputerScience and Engineering
Department of ComputerScience
4/4
Part 2. Binary Tree
Required Questions
Question 8.
For each of the following key sequences ... recursive_Insert
Faculty of ComputerScience and Engineering
Department of ComputerScience
2/4
Return element of s is appended into q with the same order. For
example if q = {1,2,3}, ... (compute(a,n)>compute(a,n-1))?compute(a,n):compute(a,n-1)
3233
6,5,4
4,5,6
Faculty of ComputerScience and Engineering
Department of ComputerScience
1/4
DATA STRUCTURES & ALGORITHMS
Tutorial 3 Questions...
...
Faculty of ComputerScience and Engineering
Department of ComputerScience
2/3
Question 6.
Suggest a data structure that supports ... for the problem that follow:
public class BinaryTree <E extends Comparable<E>> {
private class Node {
E data;
Node left, right;
}
Node root;
}
Faculty of ComputerScience ... left, right;
}
Node root;
}
Faculty of ComputerScience and Engineering
Department of ComputerScience
3/3
Write a recursive method called isCompleteBinaryTree() that returns true if...