Shortest Path Algorithms
Luis Goddyn, Math 408 Given an edge weighted graph (G, d), d : E(G) and two vertices s, t V (G), the Shortest Path Problem is to nd an s, t-path P whose total weight is as small as possible. Here, G may be either directed or undir
Mathematical Induction Solutions
Mathematical Induction Solutions
Problems
, so the base case is finished.
1.(a) Base Step: 1 = 1(1+1)
2
Induction Step: Suppose the result holds for some positive integer n. We need to show
the result holds for n + 1. We h
II.b Double Integrals in (x, y) Coordinates
In the previous section we encountered iterated integrals. We first practice
working these out, and later consider applications.
Example 1.
Evaluate the iterated integral
Z
Answer.
2
0
Z
y
cos(y 2 ) dx dy.
0
Thi
Answers to Proof Homework sheet.
1) If a, b are integers, The product is odd if and only if a and b are both odd.
Proof:
Part 1) If a and b are both odd then ab is odd.
Proof: By definition a = 2n + 1 and b = 2m + 1 for n, m integers. Now consider the
pro
Chapter 8 Characters and Strings
Principle of enumeration
Computers tend to be good at working with numeric
data.
The ability to represent an integer value, however,
also makes it easy to work with other data types as
long as it is possible to represent
Translation in Predicate Logic Identity
Identity Identity is a special two place relation in Predicate Logic. = is a logical
symbol meaning it always has the same interpretation namely, identity. This is unlike P
or Q which can change meaning with context
Finding Mathematical Proofs:
First, carefully read the statement you want to prove. Do you understand
all the notation? Rewrite the statement in your own words. Break it down and
consider each part separately. Sometimes the parts of a multipart statement
JHU-CTY Theory of Computation (TCOM) Lancaster 2007 ~ Instructors Kayla Jacobs & Adam Groce
PROOF BY INDUCTION PROBLEMS
* (1) Some Sum Facts
Prove the following identities using induction.
a) Sum of consecutive odd natural numbers:
1 + 3 + 5 + + 2(n-1)-1)
TSP Lower Bounds
Luis Goddyn, Math 408 We give here two techniques for obtaining lower bounds on TSP instances. That is, for a given instance (V, d), we would like to find the largest possible number t such that (T ) t for every TSP tour T . We first deri
TSP Heuristics
Luis Goddyn, Math 408 There are two types of heuristic methods for finding optimal TSP tours. Scratch methods produce an initial tour which is hopefully fairly close (say 10%) from being optimal. Improvement methods (sometimes called post-o
Some Stu about Polytopes, with an Application
Luis Goddyn, Math 408
1
The Convex Hull
Let V be a set of points in Euclidean d-space n . Intuitively, the convex hull of V , conv.hull(V ), is the set of points within a balloon which has been stretched aroun
Sequential Circuits
Chapter 4
S. Dandamudi
Outline
Introduction
Clock signal
Example chips
Example sequential circuits
Propagation delay
Shift registers
Counters
Latches
SR latch
Clocked SR latch
D latch
JK latch
Sequential circuit design
Simple