Practical Discrete Mathematics

Kup polskie wydanie:

Spis treści książki

• Practical Discrete Mathematics
• Why subscribe?
• Contributors
• About the authors
• About the reviewer
• Packt is searching for authors like you
• Preface
  • Who this book is for
  • What this book covers
    • Part I Basic Concepts of Discrete Math
    • Part II Implementing Discrete Mathematics in Data and Computer Science
    • Part III Real-World Applications of Discrete Mathematics
  • To get the most out of this book
  • Download the example code files
  • Download the color images
  • Conventions used
  • Get in touch
  • Reviews
• Part I Basic Concepts of Discrete Math
• Chapter 1: Key Concepts, Notation, Set Theory, Relations, and Functions
  • What is discrete mathematics?
  • Elementary set theory
    • DefinitionSets and set notation
    • Definition: Elements of sets
    • Definition: The empty set
    • Example: Some examples of sets
    • Definition: Subsets and supersets
    • Definition: Set-builder notation
    • Example: Using set-builder notation
    • Definition: Basic set operations
    • Definition: Disjoint sets
    • Example: Even and odd numbers
    • Theorem: De Morgans laws
    • Example: De Morgan's Law
    • Definition: Cardinality
    • Example: Cardinality
  • Functions and relations
    • Definition: Relations, domains, and ranges
    • Definition: Functions
    • Examples: Relations versus functions
    • Example: Functions in elementary algebra
    • Example: Python functions versus mathematical functions
  • Summary
• Chapter 2: Formal Logic and Constructing Mathematical Proofs
  • Formal Logic and Proofs by Truth Tables
    • Basic Terminology for Formal Logic
    • Example an invalid argument
    • Example all penguins live in South Africa!
    • Cores Ideas in Formal Logic
    • Truth Tables
    • Example The Converse
    • Example Transitivity Law of Conditional Logic
    • Example De Morgan's Laws
    • Example The Contrapositive
  • Direct Mathematical Proofs
    • Example Products of Even and Odd Integers
    • Example roots of even numbers
    • Shortcut The Contrapositive
  • Proof by Contradiction
    • Example is there a smallest positive rational number?
    • Example Prove is an Irrational Number
    • Example How Many Prime Numbers Are There?
  • Proof by mathematical induction
    • Example Adding 1 + 2 + + n
    • Example Space-Filling Shapes
    • Example exponential versus factorial growth
  • Summary
• Chapter 3: Computing with Base-n Numbers
  • Understanding base-n numbers
    • Example Decimal numbers
    • Definition Base-n numbers
  • Converting between bases
    • Converting base-n numbers to decimal numbers
    • Example Decimal value of a base-6 number
    • Base-n to decimal conversion
    • Example Decimal to base-2 (binary) conversion
    • Example Decimal to binary and hexadecimal conversions in Python
  • Binary numbers and their applications
    • Boolean algebra
    • Example Netflix users
  • Hexadecimal numbers and their application
    • Example Defining locations in computer memory
    • Example Displaying error messages
    • Example Media Access Control (MAC) addresses
    • Example Defining colors on the web
  • Summary
• Chapter 4: Combinatorics Using SciPy
  • The fundamental counting rule
    • Definition the Cartesian product
    • Theorem the cardinality of Cartesian products of finite sets
    • Definition the Cartesian product (for n sets)
    • Theorem the fundamental counting rule
    • Example bytes
    • Example colors on computers
  • Counting permutations and combinations of objects
    • Definition permutation
    • Example permutations of a simple set
    • Theorem permutations of a set
    • Example playlists
    • Growth of factorials
    • Theorem k-permutations of a set
    • Definition combination
    • Example combinations versus permutation for a simple set
    • Theorem combinations of a set
    • Binomial coefficients
    • Example teambuilding
    • Example combinations of balls
  • Applications to memory allocation
    • Example pre-allocating memory
  • Efficacy of brute-force algorithms
    • Example Caesar cipher
    • Example the traveling salesman problem
  • Summary
• Chapter 5: Elements of Discrete Probability
  • The basics of discrete probability
    • Definition random experiment
    • Definitions outcomes, events, and sample spaces
    • Example tossing coins
    • Example tossing multiple coins
    • Definition probability measure
    • Theorem elementary properties of probability
    • Example sports
    • Theorem Monotonicity
    • Theorem Principle of Inclusion-Exclusion
    • Definition Laplacian probability
    • Theorem calculating Laplacian probabilities
    • Example tossing multiple coins
    • Definition independent events
    • Example tossing many coins
  • Conditional probability and Bayes' theorem
    • Definition conditional probability
    • Example temperatures and precipitation
    • Theorem multiplication rules
    • Theorem the Law of Total Probability
    • Theorem Bayes' theorem
  • Bayesian spam filtering
  • Random variables, means, and variance
    • Definition random variable
    • Example data transfer errors
    • Example empirical random variable
    • Definition expectation
    • Example empirical random variable
    • Definition variance and standard deviation
    • Theorem practical calculation of variance
    • Example empirical random variable
  • Google PageRank I
  • Summary
• Part II Implementing Discrete Mathematics in Data and Computer Science
• Chapter 6: Computational Algorithms in Linear Algebra
  • Understanding linear systems of equations
    • Definition Linear equations in two variables
    • Definition The Cartesian coordinate plane
    • Example A linear equation
    • Definition System of two linear equations in two variables
    • Definition Systems of linear equations and their solutions
    • Definition Consistent, inconsistent, and dependent systems
  • Matrices and matrix representations of linear systems
    • Definition Matrices and vectors
    • Definition Matrix addition and subtraction
    • Definition Scalar multiplication
    • Definition Transpose of a matrix
    • Definition Dot product of vectors
    • Definition Matrix multiplication
    • Example Multiplying matrices by hand and with NumPy
  • Solving small linear systems with Gaussian elimination
    • Definition Leading coefficient (pivot)
    • Definition Reduced row echelon form
    • Algorithm Gaussian elimination
    • Example 3-by-3 linear system
  • Solving large linear systems with NumPy
    • Example A 3-by-3 linear system (with NumPy)
    • Example Inconsistent and dependent systems with NumPy
    • Example A 10-by-10 linear system (with NumPy)
  • Summary
• Chapter 7: Computational Requirements for Algorithms
  • Computational complexity of algorithms
  • Understanding Big-O Notation
  • Complexity of algorithms with fundamental control structures
    • Sequential flow
    • Selection flow
    • Repetitive flow
  • Complexity of common search algorithms
    • Linear search algorithm
    • Binary search algorithm
  • Common classes of computational complexity
  • Summary
  • References
• Chapter 8: Storage and Feature Extraction of Graphs, Trees, and Networks
  • Understanding graphs, trees, and networks
    • Definition: graph
    • Definition: degree of a vertex
    • Definition: paths
    • Definition: cycles
    • Definition: trees or acyclic graphs
    • Definition: networks
    • Definition: directed graphs
    • Definition: directed networks
    • Definition: adjacent vertices
    • Definition: connected graphs and connected components
  • Using graphs, trees, and networks
  • Storage of graphs and networks
    • Definition: adjacency list
    • Definition: adjacency matrix
    • Definition: adjacency matrix for a directed graph
    • Efficient storage of adjacency data
    • Definition: weight matrix of a network
    • Definition: weight matrix of a directed network
  • Feature extraction of graphs
    • Degrees of vertices in a graph
    • The number of paths between vertices of a specified length
    • Theorem: powers of adjacency matrices
    • Matrix powers in Python
    • Theorem: minimum-edge paths between vi and vj
  • Summary
• Chapter 9: Searching Data Structures and Finding Shortest Paths
  • Searching Graph and Tree data structures
  • Depth-first search (DFS)
    • A Python implementation of DFS
  • The shortest path problem and variations of the problem
    • Shortest paths on networks
    • Beyond Shortest-Distance Paths
    • Shortest Path Problem Statement
    • Checking whether Solutions Exist
  • Finding Shortest Paths with Brute Force
  • Dijkstra's Algorithm for Finding Shortest Paths
    • Dijkstra's algorithm
    • Applying Dijkstra's Algorithm to a Small Problem
  • Python Implementation of Dijkstra's Algorithm
    • Example shortest paths
    • Example A network that is not connected
  • Summary
• Part III Real-World Applications of Discrete Mathematics
• Chapter 10: Regression Analysis with NumPy and Scikit-Learn
  • Dataset
  • Best-fit lines and the least-squares method
    • Variable
    • Linear relationship
    • Regression
    • The line of best fit
    • The least-squares method and the sum of squared errors
  • Least-squares lines with NumPy
  • Least-squares curves with NumPy and SciPy
  • Least-squares surfaces with NumPy and SciPy
  • Summary
• Chapter 11: Web Searches with PageRank
  • The Development of Search Engines over time
  • Google PageRank II
  • Implementing the PageRank algorithm in Python
  • Applying the Algorithm to Real Data
  • Summary
• Chapter 12: Principal Component Analysis with Scikit-Learn
  • Understanding eigenvalues, eigenvectors, and orthogonal bases
  • The principal component analysis approach to dimensionality reduction
  • The scikit-learn implementation of PCA
  • An application to real-world data
  • Summary
• Other Books You May Enjoy
  • Leave a review - let other readers know what you think

pokaż cały spis treści

ukryj recenzje