Paul Williams

Topics

• Taxonomy: mathcentre
  • 1 - Numbers and Computation
    • 1.1 - Number Concepts
      • 1.1.1 - Natural
      • 1.1.2 - Integers
      • 1.1.3 - Rational
      • 1.1.4 - Irrational
      • 1.1.5 - Algebraic
      • 1.1.6 - Real
      • 1.1.7 - Complex
      • 1.1.8 - Famous Numbers
        • 1.1.8.1 - 0
        • 1.1.8.2 - pi
        • 1.1.8.3 - e
        • 1.1.8.4 - i
        • 1.1.8.5 - Golden Mean
      • 1.1.9 - Symbols, constants
    • 1.2 - Arithmetic
      • 1.2.1 - Operations
        • 1.2.1.1 - Addition
        • 1.2.1.2 - Subtraction
        • 1.2.1.3 - Multiplication
        • 1.2.1.4 - Division
        • 1.2.1.5 - Roots
        • 1.2.1.6 - Factorials
        • 1.2.1.7 - Factoring
        • 1.2.1.8 - Properties of Operations
        • 1.2.1.9 - Estimation
      • 1.2.2 - Fractions
        • 1.2.2.1 - Addition
        • 1.2.2.2 - Subtraction
        • 1.2.2.3 - Multiplication
        • 1.2.2.4 - Division
        • 1.2.2.5 - Ratio and Proportion
        • 1.2.2.6 - Equivalent Fractions
      • 1.2.3 - Decimals
        • 1.2.3.1 - Addition
        • 1.2.3.2 - Subtraction
        • 1.2.3.3 - Multiplication
        • 1.2.3.4 - Division
        • 1.2.3.5 - Percents
      • 1.2.4 - Comparison of numbers
      • 1.2.5 - Exponents
        • 1.2.5.1 - Multiplication
        • 1.2.5.2 - Division
        • 1.2.5.3 - Powers
        • 1.2.5.4 - Integer Exponents
        • 1.2.5.5 - Rational Exponents
      • 1.2.6 - Logarithms
        • 1.2.6.1 - Addition
        • 1.2.6.2 - Subtraction
        • 1.2.6.3 - Multiplication
        • 1.2.6.4 - Natural logarithms
        • 1.2.6.5 - Change of base
    • 1.3 - Patterns and Sequences
      • 1.3.1 - Number Patterns
      • 1.3.2 - Fibonacci Sequence
      • 1.3.3 - Arithmetic Sequence
      • 1.3.4 - Geometric Sequence
    • 1.4 - Measurement
      • 1.4.1 - Units of Measurement
        • 1.4.1.1 - Metric System
        • 1.4.1.2 - Standard Units
        • 1.4.1.3 - Nonstandard Units
      • 1.4.2 - Linear Measure
        • 1.4.2.1 - Distance
        • 1.4.2.2 - Circumference
        • 1.4.2.3 - Perimeter
      • 1.4.3 - Area
        • 1.4.3.1 - Area of Polygons
        • 1.4.3.2 - Area of Circles
        • 1.4.3.3 - Surface Area
        • 1.4.3.4 - Nonstandard Shapes
      • 1.4.4 - Volume
      • 1.4.5 - Weight and Mass
      • 1.4.6 - Temperature
      • 1.4.7 - Time
      • 1.4.8 - Speed
      • 1.4.9 - Money
      • 1.4.10 - Scale
  • 2 - Logic and Foundations
    • 2.1 - Logic
      • 2.1.1 - Logic
      • 2.1.2 - Propositional and Predicate Logic
      • 2.1.3 - Methods of Proof
    • 2.2 - Set Theory
      • 2.2.1 - Sets and Set Operations
      • 2.2.2 - Relations and Functions
      • 2.2.3 - Cardinality
      • 2.2.4 - Axiom of Choice
    • 2.3 - Computability and Decidability
    • 2.4 - Model Theory
  • 3 - Algebra and Number Theory
    • 3.1 - Algebra
      • 3.1.1 - Graphing Techniques
      • 3.1.2 - Algebraic Manipulation
        • 3.1.2.1 - Collecting like terms
        • 3.1.2.2 - Rules of indices for single term expressions
          • 3.1.2.2.1 - Surds
        • 3.1.2.3 - Polynomial expressions
          • 3.1.2.3.1 - Multiplying out brackets
            • 3.1.2.3.1.1 - Multiplying a single-term expression and bracket
            • 3.1.2.3.1.2 - Multiplying two brackets
            • 3.1.2.3.1.3 - Pure powers
            • 3.1.2.3.1.4 - The difference of two squares
            • 3.1.2.3.1.5 - Multiple brackets
          • 3.1.2.3.2 - Factoring polynomials
            • 3.1.2.3.2.1 - Single term common factor
            • 3.1.2.3.2.2 - Quadratic expressions
            • 3.1.2.3.2.3 - Higher powers
            • 3.1.2.3.2.4 - Multiple variable expressions
        • 3.1.2.4 - Algebraic fractions
          • 3.1.2.4.1 - Cancelling
            • 3.1.2.4.1.1 - Cancelling fractions involving factorising
          • 3.1.2.4.2 - Multiplying by -1
          • 3.1.2.4.3 - Adding and subtracting algebraic fractions
          • 3.1.2.4.4 - Multiplying and dividing algebraic fractions
          • 3.1.2.4.5 - Partial fractions
          • 3.1.2.4.6 - Rationalizing surd expressions
        • 3.1.2.5 - Completing the square of a quadratic expression
        • 3.1.2.6 - Rearranging equations
          • 3.1.2.6.1 - Substituting from one formula into another
          • 3.1.2.6.2 - Making a specified letter the subject of a formula
          • 3.1.2.6.3 - Substitution
        • 3.1.2.7 - De-nesting surds
      • 3.1.3 - Functions
        • 3.1.3.1 - Linear
        • 3.1.3.2 - Quadratic
        • 3.1.3.3 - Polynomial
        • 3.1.3.4 - Rational
        • 3.1.3.5 - Exponential
        • 3.1.3.6 - Logarithmic
        • 3.1.3.7 - Piece-wise
        • 3.1.3.8 - Step
        • 3.1.3.9 - Hyperbolics
          • 3.1.3.9.1 - Hyperbolic functions
          • 3.1.3.9.2 - Inverse hyperbolic functions
          • 3.1.3.9.3 - Hyperbolic identities
      • 3.1.4 - Equations
        • 3.1.4.1 - Linear
        • 3.1.4.2 - Quadratic
        • 3.1.4.3 - Polynomial
        • 3.1.4.4 - Rational
        • 3.1.4.5 - Exponential
        • 3.1.4.6 - Logarithmic
        • 3.1.4.7 - Systems
      • 3.1.5 - Inequalities
      • 3.1.6 - Matrices
      • 3.1.7 - Sequences and Series
      • 3.1.8 - Algebraic Proof
    • 3.2 - Linear Algebra
      • 3.2.1 - Systems of Linear Equations
      • 3.2.2 - Matrix algebra
      • 3.2.3 - Vectors in R3
      • 3.2.4 - Vector Spaces
      • 3.2.5 - Linear Transformations
      • 3.2.6 - Eigenvalues and Eigenvectors
      • 3.2.7 - Inner Product Spaces
    • 3.3 - Abstract Algebra
      • 3.3.1 - Groups
      • 3.3.2 - Rings and Ideals
      • 3.3.3 - Fields
      • 3.3.4 - Galois Theory
      • 3.3.5 - Multilinear Algebra
    • 3.4 - Number Theory
      • 3.4.1 - Integers
      • 3.4.2 - Primes
        • 3.4.2.1 - Divisibility
        • 3.4.2.2 - Factorization
        • 3.4.2.3 - Distributions of Primes
      • 3.4.3 - Congruences
      • 3.4.4 - Diophantine Equations
      • 3.4.5 - Irrational Numbers
      • 3.4.6 - Famous Problems
      • 3.4.7 - Coding Theory
      • 3.4.8 - Cryptography
    • 3.5 - Category Theory
    • 3.6 - K-Theory
    • 3.7 - Homological Algebra
    • 3.8 - Modular Arithmetic
  • 4 - Discrete Mathematics
    • 4.1 - Cellular Automata
    • 4.2 - Combinatorics
      • 4.2.1 - Combinations
      • 4.2.2 - Permutations
    • 4.3 - Game Theory
    • 4.4 - Algorithms
    • 4.5 - Recursion
    • 4.6 - Graph Theory
    • 4.7 - Linear Programming
    • 4.8 - Order and Lattices
    • 4.9 - Theory of Computation
    • 4.10 - Chaos
  • 5 - Geometry and Topology
    • 5.1 - Geometric Proof
    • 5.2 - Plane Geometry
      • 5.2.1 - Measurement
      • 5.2.2 - Lines and Planes
      • 5.2.3 - Angles
      • 5.2.4 - Triangles
        • 5.2.4.1 - Properties
        • 5.2.4.2 - Congruence
        • 5.2.4.3 - Similarity
        • 5.2.4.4 - Pythagorean Theorem
      • 5.2.5 - Polygons
        • 5.2.5.1 - Properties
        • 5.2.5.2 - Regular
        • 5.2.5.3 - Irregular
        • 5.2.5.4 - Congruence
        • 5.2.5.5 - Similarity
      • 5.2.6 - Circles
      • 5.2.7 - Patterns
        • 5.2.7.1 - Geometric Patterns
        • 5.2.7.2 - Tilings and Tessellations
        • 5.2.7.3 - Symmetry
        • 5.2.7.4 - Golden Ratio
      • 5.2.8 - Transformations
        • 5.2.8.1 - Translation
        • 5.2.8.2 - Rotation
        • 5.2.8.3 - Reflection
        • 5.2.8.4 - Scaling
    • 5.3 - Solid Geometry
      • 5.3.1 - Dihedral Angles
      • 5.3.2 - Spheres
      • 5.3.3 - Cones
      • 5.3.4 - Cylinders
      • 5.3.5 - Pyramids
      • 5.3.6 - Prisms
      • 5.3.7 - Polyhedra
    • 5.4 - Analytic Geometry
      • 5.4.1 - Cartesian Coordinates
      • 5.4.2 - Lines
      • 5.4.3 - Circles
      • 5.4.4 - Planes
      • 5.4.5 - Conics
      • 5.4.6 - Polar Coordinates
      • 5.4.7 - Parametric Curves
      • 5.4.8 - Surfaces
      • 5.4.9 - Distance Formula
    • 5.5 - Projective Geometry
    • 5.6 - Differential Geometry
    • 5.7 - Algebraic Geometry
    • 5.8 - Topology
      • 5.8.1 - Point Set Topology
      • 5.8.2 - General Topology
      • 5.8.3 - Differential Topology
      • 5.8.4 - Algebraic Topology
    • 5.9 - Trigonometry
      • 5.9.1 - Angles
      • 5.9.2 - Trigonometric Functions
      • 5.9.3 - Inverse Trigonometric Functions
      • 5.9.4 - Trigonometric Identities
      • 5.9.5 - Trigonometric Equations
      • 5.9.6 - Roots of Unity
      • 5.9.7 - Spherical Trigonometry
    • 5.10 - Fractal Geometry
  • 6 - Calculus
    • 6.1 - Single Variable
      • 6.1.1 - Functions
      • 6.1.2 - Limits
      • 6.1.3 - Continuity
      • 6.1.4 - Differentiation
      • 6.1.5 - Integration
      • 6.1.6 - Series
    • 6.2 - Several Variables
      • 6.2.1 - Functions of Several Variables
      • 6.2.2 - Limits
      • 6.2.3 - Continuity
      • 6.2.4 - Partial Derivatives
      • 6.2.5 - Multiple integrals
      • 6.2.6 - Taylor Series
    • 6.3 - Advanced Calculus
      • 6.3.1 - Vector Valued Functions
      • 6.3.2 - Line Integrals
      • 6.3.3 - Surface Integrals
      • 6.3.4 - Stokes Theorem
      • 6.3.5 - Curvilinear Coordinates
      • 6.3.6 - Linear spaces
      • 6.3.7 - Fourier Series
      • 6.3.8 - Orthogonal Functions
    • 6.4 - Tensor Calculus
    • 6.5 - Calculus of Variations
    • 6.6 - Operational Calculus
  • 7 - Analysis
    • 7.1 - Real Analysis
      • 7.1.1 - Metric Spaces
      • 7.1.2 - Convergence
      • 7.1.3 - Continuity
      • 7.1.4 - Differentiation
      • 7.1.5 - Integration
      • 7.1.6 - Measure Theory
    • 7.2 - Complex Analysis
      • 7.2.1 - Convergence
      • 7.2.2 - Infinite Series
      • 7.2.3 - Analytic Functions
      • 7.2.4 - Integration
      • 7.2.5 - Contour Integrals
      • 7.2.6 - Conformal Mappings
      • 7.2.7 - Several Complex Variables
    • 7.3 - Numerical Analysis
      • 7.3.1 - Computer Arithmetic
      • 7.3.2 - Solutions of Equations
      • 7.3.3 - Solutions of Systems
      • 7.3.4 - Interpolation
      • 7.3.5 - Numerical Differentiation
      • 7.3.6 - Numerical Integration
      • 7.3.7 - Numerical Solutions of ODEs
      • 7.3.8 - Numerical Solutions of PDEs
    • 7.4 - Integral Transformations
      • 7.4.1 - Fourier Transforms
      • 7.4.2 - Laplace Transforms
      • 7.4.3 - Hankel Transforms
      • 7.4.4 - Wavelets
      • 7.4.5 - Other Transforms
    • 7.5 - Signal Analysis
      • 7.5.1 - Sampling Theory
      • 7.5.2 - Filters
      • 7.5.3 - Noise
      • 7.5.4 - Data Compression
      • 7.5.5 - Image Processing
    • 7.6 - Functional Analysis
      • 7.6.1 - Hilbert Spaces
      • 7.6.2 - Banach Spaces
      • 7.6.3 - Topological Spaces
      • 7.6.4 - Locally Convex Spaces
      • 7.6.5 - Bounded Operators
      • 7.6.6 - Spectral Theorem
      • 7.6.7 - Unbounded Operators
    • 7.7 - Harmonic Analysis
    • 7.8 - Global Analysis
  • 8 - Differential and Difference Equations
    • 8.1 - Ordinary Differential Equations
      • 8.1.1 - First Order
      • 8.1.2 - Second Order
      • 8.1.3 - Linear Oscillations
      • 8.1.4 - Nonlinear Oscillations
      • 8.1.5 - Systems of Differential Equations
      • 8.1.6 - Sturm Liouville Problems
      • 8.1.7 - Special Functions
      • 8.1.8 - Power Series Methods
      • 8.1.9 - Laplace Transforms
    • 8.2 - Partial Differential Equations
      • 8.2.1 - First Order
      • 8.2.2 - Elliptic
      • 8.2.3 - Parabolic
      • 8.2.4 - Hyperbolic
      • 8.2.5 - Integral Transforms
      • 8.2.6 - Integral Equations
      • 8.2.7 - Potential Theory
      • 8.2.8 - Nonlinear Equations
      • 8.2.9 - Symmetries and Integrability
    • 8.3 - Difference Equations
      • 8.3.1 - First Order
      • 8.3.2 - Second Order
      • 8.3.3 - Linear Systems
      • 8.3.4 - Z Transforms
      • 8.3.5 - Orthogonal Polynomials
    • 8.4 - Dynamical Systems
      • 8.4.1 - 1D Maps
      • 8.4.2 - 2D Maps
      • 8.4.3 - Lyapunov Exponents
      • 8.4.4 - Bifurcations
      • 8.4.5 - Fractals
      • 8.4.6 - Differentiable Dynamics
      • 8.4.7 - Conservative Dynamics
      • 8.4.8 - Chaos
      • 8.4.9 - Complex Dynamical Systems
  • 9 - Statistics and Probability
    • 9.1. - Data Collection
      • 9.1.1. - Experimental Design
      • 9.1.2. - Sampling and Surveys
      • 9.1.3. - Data and Measurement Issues
    • 9.2. - Data Summary and Presentation
      • 9.2.1. - Summary Statistics
        • 9.2.1.1. - Measures of Central Tendency
        • 9.2.1.2. - Measures of Spread
      • 9.2.2. - Data Representation
        • 9.2.2.1. - Graphs and Plots
        • 9.2.2.2. - Tables
    • 9.3. - Statistical Inference and Techniques
      • 9.3.1. - Sampling Distributions
      • 9.3.2. - Regression and Correlation
      • 9.3.3. - Confidence Intervals
      • 9.3.4. - Hypothesis Tests
      • 9.3.5. - Statistical Quality Control
      • 9.3.6. - Non-parametric Techniques
      • 9.3.7. - Multivariate Techniques
      • 9.3.8. - Survival Analysis
      • 9.3.9. - Bayesian Statistics
    • 9.4. - Probability
      • 9.4.3 - Limit Theorems
        • 9.4.3.1 - Central Limit Theorem
        • 9.4.3.2 - Law of Large Numbers
      • 9.4.5 - Stochastic Processes
        • 9.4.5.1 - Brownian Motion
        • 9.4.5.2 - Markov Chains
        • 9.4.5.3 - Queuing Theory
      • 9.4.6 - Probability Measures
      • 9.4.7 - Simulation
      • 9.4.1. - Elementary Probability
        • 9.4.1.1. - Sample Space and Events
        • 9.4.1.2. - General Rules
        • 9.4.1.3. - Combinations and Permutations
        • 9.4.1.4. - Random Variables
      • 9.4.2. - Univariate Distributions
        • 9.4.2.1. - Discrete Distributions
        • 9.4.2.2. - Continuous Distributions
        • 9.4.2.3. - Expected Value
      • 9.4.4. - Multivariate Distributions
        • 9.4.4.1 - Joint
        • 9.4.4.2 - Conditional
        • 9.4.4.3 - Expectations
  • 10.0 - Applied Mathematics
    • 10.1 - Mathematical Physics
    • 10.2 - Mathematical Economics
    • 10.3 - Mathematical Biology
    • 10.4 - Mathematics for Business
    • 10.5 - Engineering Mathematics
      • 10.5.1 - Mathematics for Materials
    • 10.6 - Mathematical Sociology
    • 10.7 - Mathematics for Social Sciences
      • 10.7.1 - Mathematics for Social Policy and Social Work
    • 10.8 - Mathematics for Computer Science
    • 10.9 - Mathematics for Humanities
      • 10.9.1 - Mathematics for Art, Design and Communication
      • 10.9.2 - Mathematics for Built Environment
      • 10.9.3 - Mathematics for History, Classics and Archaeology
      • 10.9.4 - Mathematics for Law
      • 10.9.5 - Mathematics for Performing Arts
      • 10.9.6 - Mathematics for Education
      • 10.9.7 - Mathematics for Languages, Linguistics and Area Studies
        • 10.9.7.1 - Mathematics for English
      • 10.9.8 - Mathematics for Philosophical and Religious Studies
    • 10.10 - Consumer Mathematics
    • 10.11 - Mathematics for Economics
    • 10.12 - Mathematics for Life Sciences
      • 10.12.1 - Mathematics for Geography, Earth and Environmental Sciences
      • 10.12.2 - Mathematics for Bioscience
      • 10.12.3 - Mathematics for Health Sciences and Practice
      • 10.12.4 - Mathematics for Medicine, Dentistry and Veterinary Medicine
    • 10.13 - Mathematics for Psychology
    • 10.14 - Mathematics for Physical Sciences
    • 10.15 - Mathematics for Sociology, Anthropology and Politics
    • 10.16 - Mathematics for Hospitality, Leisure, Sport and Tourism
  • 11.0 - Mathematics History
    • 11.1 - General
    • 11.2 - Famous Problems
    • 11.3 - Biographies of Mathematicians
  • 12.0 - Mathematical Education
• Taxonomy: Kind of activity
  • 1 - Create
    • 1.1 - Give an example
    • 1.2 - Give a starting point
    • 1.3 - Give an argument
  • 2 - Evaluate
    • 2.1 - Make a decision
    • 2.2 - Classify an object
  • 3 - Analyze
    • 3.1 - Interpret a situation
    • 3.2 - Select from a list of options
    • 3.3 - Identify an error
  • 4 - Apply
    • 4.1 - Use formulas
    • 4.2 - Perform a routine calculation or algorithm
    • 4.3 - Apply an algorithm
  • 5 - Understand
    • 5.1 - Use an appropriate formula
    • 5.2 - Identify a component
  • 6 - Remember
    • 6.1 - Recall a fact
• Taxonomy: Context
  • 1 - Mathematics
    • 1.1 - Pure
    • 1.2 - Applied
    • 1.3 - For...
      • 1.3.1 - Accountancy
      • 1.3.2 - Business
      • 1.3.3 - Psychology
      • 1.3.4 - Science
        • 1.3.4.1 - Biomedicine
        • 1.3.4.2 - Chemistry
        • 1.3.4.3 - Biology
  • 2 - Statistics
    • 2.1 - Bayesian
    • 2.2 - Frequentist
  • 3 - Physics
    • 3.1 - Theoretical
  • 4 - Engineering
    • 4.1 - Civil
    • 4.2 - Electrical
    • 4.3 - Chemical
  • 5 - Chemistry