Geometry
Main concepts
Length, area, and volume


Geometry
Main concepts
Length, area, and volume
Metrics and measures


Geometry
Main concepts
Congruence and similarity


Geometry
Main concepts
Compass and straightedge constructions


Euclidean geometry
Differential geometry
Topology
Algebraic geometry
Complex geometry
Discrete geometry
Computational geometry
Geometric group theory
Convex geometry


Geometry
Contemporary geometry
Euclidean geometry


Geometry
Contemporary geometry
Differential geometry


Geometry
Contemporary geometry
Differential geometry
Non-Euclidean geometry


Geometry
Contemporary geometry
Algebraic geometry


Geometry
Contemporary geometry
Complex geometry


Geometry
Contemporary geometry
Discrete geometry


Geometry
Contemporary geometry
Computational geometry


Geometry
Contemporary geometry
Geometric group theory


Geometry
Contemporary geometry
Convex geometry


Art
Architecture
Physics
Other fields of mathematics


Geometry
Applications
Other fields of mathematics


Standard equation
Parameterization
Volume
Surface area
Plane sections
Pins-and-string construction
In general position
Applications
In higher dimensions


Ellipsoid
Surface area
Approximate formula


Determining the ellipse of a plane section


Steps of the construction
Semi-axes
Converse
Confocal ellipsoids
Limit case, ellipsoid of revolution
Properties of the focal hyperbola
Property of the focal ellipse


Ellipsoid
Pins-and-string construction
Steps of the construction


Ellipsoid
Pins-and-string construction
Semi-axes


Ellipsoid
Pins-and-string construction
Converse


Ellipsoid
Pins-and-string construction
Confocal ellipsoids


Ellipsoid
Pins-and-string construction
Limit case, ellipsoid of revolution


Ellipsoid
Pins-and-string construction
Properties of the focal hyperbola


Ellipsoid
Pins-and-string construction
Property of the focal ellipse


Ellipsoid
In general position
As a quadric


Ellipsoid
In general position
Parametric representation


Dynamical properties
Fluid dynamics
In probability and statistics


Ellipsoid
Applications
Dynamical properties


Ellipsoid
Applications
In probability and statistics


Connection to other algebraic structures


As partially ordered set
As algebraic structure
Connection between the two definitions


Lattice (order)
Definition
As partially ordered set


Lattice (order)
Definition
As algebraic structure


Lattice (order)
Definition
Connection between the two definitions


Lattice (order)
Examples of non-lattices


Completeness
Conditional completeness
Distributivity
Modularity
Semimodularity
Continuity and algebraicity
Complements and pseudo-complements
Jordan–Dedekind chain condition
Graded/ranked


Lattice (order)
Properties of lattices
Completeness


Lattice (order)
Properties of lattices
Conditional completeness


Lattice (order)
Properties of lattices
Distributivity


Lattice (order)
Properties of lattices
Modularity


Lattice (order)
Properties of lattices
Semimodularity


Lattice (order)
Properties of lattices
Continuity and algebraicity


Lattice (order)
Properties of lattices
Complements and pseudo-complements


Lattice (order)
Properties of lattices
Jordan–Dedekind chain condition


Lattice (order)
Properties of lattices
Graded/ranked


Lattice (order)
Important lattice-theoretic notions


Lattices in two dimensions: detailed discussion


Lattice (group)
Symmetry considerations and examples


Lattice (group)
Dividing space according to a lattice


Lattice (group)
Lattice points in convex sets


Lattice (group)
Computational lattice problems


Lattice (group)
Lattices in three dimensions


Lattice (group)
Lattices in complex space


Lattice (group)
Lattices in general vector spaces


Mathematical structure
Example: the real numbers


History
Museum
Scholarly journals
Series and titles
Clarendon Scholarships
Controversies


Oxford University Press
Museum
Clarendon Press


Oxford University Press
Scholarly journals


Oxford University Press
Series and titles


Oxford University Press
Clarendon Scholarships


Tehran Book Fair Controversy
Malcolm vs Oxford University 1986-1992
Closure of Poetry List
Tax Exemption Controversy
East African Bribery Scandal


Oxford University Press
Controversies
Tehran Book Fair Controversy


Oxford University Press
Controversies
Malcolm vs Oxford University 1986-1992


Oxford University Press
Controversies
Closure of Poetry List


Oxford University Press
Controversies
Tax Exemption Controversy


Oxford University Press
Controversies
East African Bribery Scandal


Definition and basic properties
Related concepts and properties
History
Examples
Linear maps and matrices
Basic constructions
Vector spaces with additional structure
Related structures
Related concepts


Vector space
Definition and basic properties


Vector space
Related concepts and properties


Second example: ordered pairs of numbers

Complex numbers and other field extensions


Vector space
Examples
Arrows in the plane


Second example: ordered pairs of numbers


Complex numbers and other field extensions


Vector space
Linear maps and matrices
Matrices


Vector space
Linear maps and matrices
Eigenvalues and eigenvectors


Subspaces and quotient spaces
Direct product and direct sum
Tensor product


Vector space
Basic constructions
Subspaces and quotient spaces


Vector space
Basic constructions
Direct product and direct sum


Vector space
Basic constructions
Tensor product


Vector space
Vector spaces with additional structure

Normed vector spaces and inner product spaces


Normed vector spaces and inner product spaces


Vector space
Vector spaces with additional structure
Topological vector spaces


Vector space
Vector spaces with additional structure
Topological vector spaces
Banach spaces


Vector space
Vector spaces with additional structure
Topological vector spaces
Hilbert spaces


Vector space
Vector spaces with additional structure
Algebras over fields


Vector bundles
Modules
Affine and projective spaces


Vector space
Related structures
Vector bundles


Vector space
Related structures
Affine and projective spaces


Introduction and definition
Examples
Submodules and homomorphisms
Types of modules
Further notions


Module (mathematics)
Introduction and definition


Module (mathematics)
Introduction and definition
Motivation


Module (mathematics)
Introduction and definition
Formal definition


Module (mathematics)
Submodules and homomorphisms


Relation to representation theory
Generalizations


Module (mathematics)
Further notions
Relation to representation theory


Module (mathematics)
Further notions
Generalizations


Definition Examples Properties Notation Operations Applications Generalizations History


As multidimensional arrays
As multilinear maps
Using tensor products
Tensors in infinite dimensions
Tensor fields


Tensor
Definition
As multidimensional arrays


Tensor
Definition
Tensors in infinite dimensions


Ricci calculus
Einstein summation convention
Penrose graphical notation
Abstract index notation
Component-free notation


Tensor
Notation
Einstein summation convention


Tensor
Notation
Penrose graphical notation


Tensor product
Contraction
Raising or lowering an index


Tensor
Operations
Raising or lowering an index


Continuum mechanics
Other examples from physics
Computer vision and optics
Machine learning


Tensor
Applications
Other examples from physics


Tensor
Applications
Computer vision and optics


Tensor products of vector spaces
Tensors in infinite dimensions
Tensor densities
Geometric objects
Spinors


Tensor
Generalizations
Tensor products of vector spaces


Tensor
Generalizations
Tensors in infinite dimensions


Tensor
Generalizations
Geometric objects


Publications (Books Only)
Notes
References
External links


Philip J. Davis
Publications (Books Only)


Abstract algebraic aspects and generalizations


Addition, scalar multiplication, and transposition


Matrix (mathematics)
Basic operations
Matrix multiplication


Matrix (mathematics)
Basic operations
Row operations


Matrix (mathematics)
Basic operations
Submatrix


Matrix (mathematics)
Linear transformations


Matrix (mathematics)
Square matrix
Main types

Diagonal and triangular matrix
Identity matrix
Symmetric or skew-symmetric matrix
Invertible matrix and its inverse
Definite matrix
Orthogonal matrix


Matrix (mathematics)
Square matrix
Main types
Diagonal and triangular matrix


Matrix (mathematics)
Square matrix
Main types
Identity matrix


Matrix (mathematics)
Square matrix
Main types
Symmetric or skew-symmetric matrix


Matrix (mathematics)
Square matrix
Main types
Invertible matrix and its inverse


Matrix (mathematics)
Square matrix
Main types
Definite matrix


Matrix (mathematics)
Square matrix
Main types
Orthogonal matrix


Matrix (mathematics)
Square matrix
Main operations

Trace
Determinant
Eigenvalues and eigenvectors


Matrix (mathematics)
Square matrix
Main operations
Trace


Matrix (mathematics)
Square matrix
Main operations
Determinant


Matrix (mathematics)
Square matrix
Main operations
Eigenvalues and eigenvectors


Matrix (mathematics)
Computational aspects


Abstract algebraic aspects and generalizations

Matrices with more general entries
Relationship to linear maps
Matrix groups
Infinite matrices
Empty matrix


Symmetries and transformations in physics


Matrix (mathematics)
Applications
Graph theory


Matrix (mathematics)
Applications
Analysis and geometry


Matrix (mathematics)
Applications
Probability theory and statistics


Matrix (mathematics)
Applications
Linear combinations of quantum states


Matrix (mathematics)
Applications
Normal modes


Matrix (mathematics)
Applications
Geometrical optics


Matrix (mathematics)
Applications
Electronics


Other historical usages of the word "matrix" in mathematics


Stanford Encyclopedia of Philosophy
Approach and history


Finding intersections of geometric objects

Spherical and nonlinear planes and their tangents


Analytic geometry
History
Ancient Greece


Analytic geometry
History
Western Europe


Cartesian coordinates (in a plane or space)


Analytic geometry
Coordinates
Polar coordinates (in a plane)


Analytic geometry
Coordinates
Cylindrical coordinates (in a space)


Analytic geometry
Coordinates
Spherical coordinates (in a space)


Lines and planes
Conic sections
Quadric surfaces


Analytic geometry
Equations and curves
Lines and planes


Analytic geometry
Equations and curves
Conic sections


Analytic geometry
Equations and curves
Quadric surfaces


Finding intersections of geometric objects


Spherical and nonlinear planes and their tangents


Vectors in Euclidean geometry
Vector spaces
Vectors in algebra
Data represented by vectors


Vector (mathematics and physics)
Vectors in Euclidean geometry


Vector (mathematics and physics)
Vector spaces


Vector (mathematics and physics)
Vectors in algebra


Vector (mathematics and physics)
Data represented by vectors


Form of recursive definitions
Examples of recursive definitions


Recursive definition
Form of recursive definitions


Recursive definition
Form of recursive definitions
Principle of recursive definition


Recursive definition
Examples of recursive definitions

Elementary functions
Prime numbers
Non-negative even numbers
Well formed formulas


Recursive definition
Examples of recursive definitions
Elementary functions


Recursive definition
Examples of recursive definitions
Prime numbers


Recursive definition
Examples of recursive definitions
Non-negative even numbers


Recursive definition
Examples of recursive definitions
Well formed formulas


Examples
Paradoxes
Classes in formal set theories
Notes
References
External links


Class (set theory)
Classes in formal set theories


Definition
Operations
Examples
Special types of binary relations
Sets versus classes
Homogeneous relation
Heterogeneous relation
Calculus of relations
Induced concept lattice
Particular relations
Preorder R\R
Fringe of a relation
Mathematical heaps


Union
Intersection
Composition
Converse
Complement
Restriction
Matrix representation


Binary relation
Operations
Matrix representation


Binary relation
Special types of binary relations


Binary relation
Particular relations
Difunctional


Binary relation
Particular relations
Ferrers type


Binary relation
Particular relations
Contact


Background
The Bourbaki collective
Working method
Works
Membership
Influence and criticism


Nicolas Bourbaki
The Bourbaki collective

Founding
World War II
Postwar until the present


Nicolas Bourbaki
The Bourbaki collective
Founding


Nicolas Bourbaki
The Bourbaki collective
World War II


Nicolas Bourbaki
The Bourbaki collective
Postwar until the present


Éléments de mathématique
Séminaire Bourbaki
Articles
La Tribu


Nicolas Bourbaki
Works
Éléments de mathématique


Nicolas Bourbaki
Works
Séminaire Bourbaki


Nicolas Bourbaki
Influence and criticism


Nicolas Bourbaki
Influence and criticism
Influence


Nicolas Bourbaki
Influence and criticism
Praise


Nicolas Bourbaki
Influence and criticism
Criticism


Definition
Examples
Unnatural isomorphism
Operations with natural transformations
Functor categories
Yoneda lemma
Historical notes


Opposite group
Abelianization
Hurewicz homomorphism
Determinant
Double dual of a vector space
Finite calculus
Tensor-hom adjunction


Natural transformation
Examples
Opposite group


Natural transformation
Examples
Abelianization


Natural transformation
Examples
Hurewicz homomorphism


Natural transformation
Examples
Determinant


Natural transformation
Examples
Double dual of a vector space


Natural transformation
Examples
Finite calculus


Natural transformation
Examples
Tensor-hom adjunction


Natural transformation
Unnatural isomorphism

Example: dual of a finite-dimensional vector space


Natural transformation
Unnatural isomorphism
Example: fundamental group of torus


Example: dual of a finite-dimensional vector space


Natural transformation
Operations with natural transformations

Vertical composition
Horizontal composition
Whiskering
Interchange law


Natural transformation
Operations with natural transformations
Vertical composition


Natural transformation
Operations with natural transformations
Horizontal composition


Natural transformation
Operations with natural transformations
Whiskering


Natural transformation
Operations with natural transformations
Interchange law


Natural transformation
Functor categories


Natural transformation
Functor categories
More examples


Motivation
Formal definition
Connection with comma categories
Examples
Properties
History


Universal property
Connection with comma categories


Tensor algebras
Products
Limits and colimits


Universal property
Examples
Tensor algebras


Universal property
Examples
Limits and colimits


Existence and uniqueness
Equivalent formulations
Relation to adjoint functors


Universal property
Properties
Existence and uniqueness


Universal property
Properties
Equivalent formulations


Universal property
Properties
Relation to adjoint functors


Synopsis
Behaviorism
Translation and meaning
Reference
Semantic ascent


Indeterminacy of translation
Analytical hypotheses


Word and Object
Translation and meaning
Indeterminacy of translation


Word and Object
Translation and meaning
Analytical hypotheses


Vagaries of reference and referential transparency


Word and Object
Reference
Acquiring reference


Word and Object
Reference
Canonical notation


Description
Examples
Diagram chasing
In higher category theory
Diagrams as functors


Commutative diagram
Description
Arrow symbols


Commutative diagram
Description
Verifying commutativity


Commutative diagram
In higher category theory


Commutative diagram
Diagrams as functors


History
How sets are defined and set notation
Membership
The empty set
Singleton sets
Subsets
Euler and Venn diagrams
Special sets of numbers in mathematics
Functions
Cardinality
Power sets
Partitions
Basic operations
Applications
Principle of inclusion and exclusion


Set (mathematics)
History
Naive set theory


Set (mathematics)
History
Axiomatic set theory


Set (mathematics)
How sets are defined and set notation

Roster notation
Semantic definition
Set-builder notation
Classifying methods of definition


Set (mathematics)
How sets are defined and set notation
Roster notation


Set (mathematics)
How sets are defined and set notation
Roster notation
Infinite sets in roster notation


Set (mathematics)
How sets are defined and set notation
Semantic definition


Set (mathematics)
How sets are defined and set notation
Set-builder notation


Set (mathematics)
How sets are defined and set notation
Classifying methods of definition


Set (mathematics)
Euler and Venn diagrams


Set (mathematics)
Special sets of numbers in mathematics


Infinite sets and infinite cardinality
The continuum hypothesis


Set (mathematics)
Cardinality
Infinite sets and infinite cardinality


Set (mathematics)
Cardinality
The continuum hypothesis


Set (mathematics)
Principle of inclusion and exclusion


History
Abstracting and indexing
Notes
References
External links


Fundamenta Mathematicae
Abstracting and indexing


History
Technical information
Uses
Legal status
Archived content legal issues
Censorship and other threats


Storage capacity and growth
Wayback Machine APIs
Website exclusion policy


Wayback Machine
Technical information
Storage capacity and growth


Wayback Machine
Technical information
Wayback Machine APIs


Wayback Machine
Technical information
Website exclusion policy


Wayback Machine
Technical information
Website exclusion policy
Oakland Archive Policy


Civil litigation
Patent law
Limitations of utility


Wayback Machine
Uses
In legal evidence
Civil litigation


Wayback Machine
Uses
In legal evidence
Patent law


Wayback Machine
Uses
In legal evidence
Limitations of utility


Wayback Machine
Archived content legal issues

Scientology
Healthcare Advocates, Inc.
Suzanne Shell
Daniel Davydiuk
FlexiSpy


Wayback Machine
Archived content legal issues
Scientology


Wayback Machine
Archived content legal issues
Healthcare Advocates, Inc.


Wayback Machine
Archived content legal issues
Suzanne Shell


Wayback Machine
Archived content legal issues
Daniel Davydiuk


Wayback Machine
Archived content legal issues
FlexiSpy


Wayback Machine
Censorship and other threats


Metamath language
Metamath proof checker
Metamath databases
Natural deduction
Other works connected to Metamath


Metamath
Metamath language
Language basics


Metamath Proof Explorer
Intuitionistic Logic Explorer
New Foundations Explorer
Higher-Order Logic Explorer
Databases without explorers
Older explorers


Metamath
Metamath databases
Metamath Proof Explorer


Metamath
Metamath databases
Intuitionistic Logic Explorer


Metamath
Metamath databases
New Foundations Explorer


Metamath
Metamath databases
Higher-Order Logic Explorer


Metamath
Metamath databases
Databases without explorers


Metamath
Metamath databases
Older explorers


Metamath
Other works connected to Metamath


Metamath
Other works connected to Metamath
Proof checkers


Metamath
Other works connected to Metamath
Editors


Absence of the Axiom of Foundation
Errata


Naive Set Theory (book)
Absence of the Axiom of Foundation


History
Values
Operations
Laws
Diagrammatic representations
Boolean algebras
Axiomatizing Boolean algebra
Propositional logic
Applications


Boolean algebra
Operations
Basic operations


Boolean algebra
Operations
Secondary operations


Monotone laws
Nonmonotone laws
Completeness
Duality principle


Boolean algebra
Diagrammatic representations


Boolean algebra
Diagrammatic representations
Venn diagrams


Boolean algebra
Diagrammatic representations
Digital logic gates


Concrete Boolean algebras
Subsets as bit vectors
The prototypical Boolean algebra
Boolean algebras: the definition
Representable Boolean algebras


Boolean algebra
Boolean algebras
Concrete Boolean algebras


Boolean algebra
Boolean algebras
Subsets as bit vectors


Boolean algebra
Boolean algebras
The prototypical Boolean algebra


Boolean algebra
Boolean algebras
Boolean algebras: the definition


Boolean algebra
Boolean algebras
Representable Boolean algebras


Boolean algebra
Axiomatizing Boolean algebra


Deductive systems for propositional logic


Boolean algebra
Propositional logic
Applications


Computers
Two-valued logic
Boolean operations


Boolean algebra
Applications
Two-valued logic


Boolean algebra
Applications
Boolean operations

Natural language
Digital logic
Naive set theory
Video cards
Modeling and CAD
Boolean searches


Boolean algebra
Applications
Boolean operations
Natural language


Boolean algebra
Applications
Boolean operations
Digital logic


Boolean algebra
Applications
Boolean operations
Naive set theory


Boolean algebra
Applications
Boolean operations
Video cards


Boolean algebra
Applications
Boolean operations
Modeling and CAD


Boolean algebra
Applications
Boolean operations
Boolean searches


Proof theory and constructive mathematics


Mathematical logic
History
Early history


Mathematical logic
History
19th century
Foundational theories


Set theory and paradoxes
Symbolic logic
Beginnings of the other branches


Mathematical logic
History
20th century
Set theory and paradoxes


Mathematical logic
History
20th century
Symbolic logic


Mathematical logic
History
20th century
Beginnings of the other branches


Mathematical logic
Formal logical systems

First-order logic
Other classical logics
Nonclassical and modal logic
Algebraic logic


Mathematical logic
Formal logical systems
First-order logic


Mathematical logic
Formal logical systems
Other classical logics


Mathematical logic
Formal logical systems
Nonclassical and modal logic


Mathematical logic
Formal logical systems
Algebraic logic


Mathematical logic
Recursion theory
Algorithmically unsolvable problems


Mathematical logic
Connections with computer science


Mathematical logic
Foundations of mathematics


Grand challenges, past and present
Topics in discrete mathematics


Discrete mathematics
Grand challenges, past and present


Discrete mathematics
Topics in discrete mathematics

Discrete analogues of continuous mathematics


Discrete mathematics
Topics in discrete mathematics
Theoretical computer science


Discrete mathematics
Topics in discrete mathematics
Information theory


Discrete mathematics
Topics in discrete mathematics
Logic


Discrete mathematics
Topics in discrete mathematics
Set theory


Discrete mathematics
Topics in discrete mathematics
Combinatorics


Discrete mathematics
Topics in discrete mathematics
Graph theory


Discrete mathematics
Topics in discrete mathematics
Number theory


Discrete mathematics
Topics in discrete mathematics
Algebraic structures


Discrete analogues of continuous mathematics

Calculus of finite differences, discrete analysis, and discrete calculus


Calculus of finite differences, discrete analysis, and discrete calculus


Defining natural language
Controlled languages
International constructed languages


Natural language
Defining natural language


Natural language
International constructed languages


History
Basic concepts and notation
Ontology
Formalized set theory
Applications
Areas of study
Objections to set theory
Set theory in mathematical education


Combinatorial set theory
Descriptive set theory
Fuzzy set theory
Inner model theory
Large cardinals
Determinacy
Forcing
Cardinal invariants
Set-theoretic topology


Set theory
Areas of study
Combinatorial set theory


Set theory
Areas of study
Descriptive set theory


Set theory
Areas of study
Fuzzy set theory


Set theory
Areas of study
Inner model theory


Set theory
Areas of study
Large cardinals


Set theory
Areas of study
Cardinal invariants


Set theory
Areas of study
Set-theoretic topology


Set theory
Set theory in mathematical education


Life
Work as a logician
Philosopher
Sense and reference
1924 diary
Personality
Important dates
Important works


Childhood (1848–69)
Studies at University (1869–74)


Gottlob Frege
Life
Studies at University (1869–74)


Logic, foundation of arithmetic
Philosophical studies
Articles on geometry


Gottlob Frege
Important works
Logic, foundation of arithmetic


Gottlob Frege
Important works
Philosophical studies


Gottlob Frege
Important works
Articles on geometry


Definition
Usage
Distinction from "if" and "only if"
In terms of Euler diagrams
More general usage


Notation
Proofs
Origin of iff and pronunciation
Usage in definitions


If and only if
Usage
Origin of iff and pronunciation


If and only if
Usage
Usage in definitions


If and only if
Distinction from "if" and "only if"


If and only if
In terms of Euler diagrams


Biography and studies
Legacy
Research
Publications


Kazimierz Kuratowski
Biography and studies

Doctoral thesis
Academic career until World War II
During and after the war


Kazimierz Kuratowski
Biography and studies
Doctoral thesis


Kazimierz Kuratowski
Biography and studies
Academic career until World War II


Kazimierz Kuratowski
Biography and studies
During and after the war


Kazimierz Kuratowski
Legacy
Kazimierz Kuratowski Prize


History
Concepts
Comparison to imperative programming
Applications


First-class and higher-order functions
Pure functions
Recursion
Strict versus non-strict evaluation
Type systems
Referential transparency
Data structures


Functional programming
Concepts
First-class and higher-order functions


Functional programming
Concepts
Pure functions


Functional programming
Concepts
Recursion


Functional programming
Concepts
Strict versus non-strict evaluation


Functional programming
Concepts
Type systems


Functional programming
Concepts
Referential transparency


Functional programming
Concepts
Data structures


Functional programming
Comparison to imperative programming

Functional programming in non-functional languages


Functional programming
Comparison to imperative programming
Imperative vs. functional programming


Functional programming
Comparison to imperative programming
Simulating state


Functional programming
Comparison to imperative programming
Efficiency issues


Functional programming in non-functional languages


Spreadsheets Academia Industry Education


Functional programming
Applications
Spreadsheets


Functional programming
Applications
Academia


Functional programming
Applications
Industry


Functional programming
Applications
Education


Multiplicity of a solution of a nonlinear system of equations


Multiplicity (mathematics)
Multiplicity of a prime factor


Multiplicity (mathematics)
Multiplicity of a root of a polynomial

Behavior of a polynomial function near a multiple root


Behavior of a polynomial function near a multiple root


Multiplicity (mathematics)
Intersection multiplicity


Multiplicity (mathematics)
In complex analysis


Definition
Properties
History
Higher mathematics
Additional applications


Addition and subtraction
Multiplication
Division


Parity (mathematics)
Properties
Addition and subtraction


Parity (mathematics)
Properties
Multiplication


Parity (mathematics)
Properties
Division


Higher dimensions and more general classes of numbers


Parity (mathematics)
Higher mathematics
Number theory


Parity (mathematics)
Higher mathematics
Group theory


Parity (mathematics)
Higher mathematics
Analysis


Parity (mathematics)
Higher mathematics
Combinatorial game theory


Parity (mathematics)
Additional applications


History and development
Legislative powers
Executive powers
Leadership roles
Selection process
Incumbency
Post-presidency
Political affiliation
Timeline of presidents


President of the United States
History and development

Origins
1789–1933
Imperial presidency
Critics of presidency's evolution


President of the United States
History and development
Origins


President of the United States
History and development
1789–1933


President of the United States
History and development
Imperial presidency


President of the United States
History and development
Critics of presidency's evolution


President of the United States
Legislative powers

Signing and vetoing bills
Setting the agenda
Promulgating regulations
Convening and adjourning Congress


President of the United States
Legislative powers
Signing and vetoing bills


President of the United States
Legislative powers
Setting the agenda


President of the United States
Legislative powers
Promulgating regulations


President of the United States
Legislative powers
Convening and adjourning Congress


President of the United States
Executive powers

Administrative powers
Foreign affairs
Commander-in-chief
Juridical powers and privileges


President of the United States
Executive powers
Administrative powers


President of the United States
Executive powers
Foreign affairs


President of the United States
Executive powers
Commander-in-chief


President of the United States
Executive powers
Juridical powers and privileges


President of the United States
Leadership roles

Head of state
Head of party
Global leader


President of the United States
Leadership roles
Head of state


President of the United States
Leadership roles
Head of party


President of the United States
Leadership roles
Global leader


President of the United States
Selection process

Eligibility
Campaigns and nomination
Election
Inauguration


President of the United States
Selection process
Eligibility


President of the United States
Selection process
Campaigns and nomination


President of the United States
Selection process
Election


President of the United States
Selection process
Inauguration


President of the United States
Incumbency

Term limit
Vacancies and succession
Declarations of inability
Removal
Circumvention of authority
Compensation
Residence
Travel
Protection


President of the United States
Incumbency
Term limit


President of the United States
Incumbency
Vacancies and succession


President of the United States
Incumbency
Declarations of inability


President of the United States
Incumbency
Removal


President of the United States
Incumbency
Circumvention of authority


President of the United States
Incumbency
Compensation


President of the United States
Incumbency
Residence


President of the United States
Incumbency
Travel


President of the United States
Incumbency
Protection


President of the United States
Post-presidency

Activities
Pension and other benefits
Presidential libraries


President of the United States
Post-presidency
Activities


President of the United States
Post-presidency
Pension and other benefits


President of the United States
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Presidential libraries


President of the United States
Political affiliation


President of the United States
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Chronological list of the oldest known living person since 1955


Oldest people
Ten oldest verified people ever


Oldest people
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Oldest people
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Oldest living people


Oldest people
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Chronological list of the oldest known living person since 1955

Chronological list of the oldest living man since 1973


Chronological list of the oldest living man since 1973


Scope of the concept
In computer programming
Examples


Relation to the axiom schema of replacement


Axiom schema of specification
Unrestricted comprehension


Axiom schema of specification
In NBG class theory


Axiom schema of specification
In higher-order settings


Axiom schema of specification
In Quine's New Foundations


Axiom schema of specification
References


Axiom schema of specification
References
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Formal statement
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Sources


Statement
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History
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Axiom schema of replacement
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Axiom schema of replacement
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Axiom schema of replacement
Relation to other axiom schemas
Collection


Axiom schema of replacement
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Separation


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Boolean rings
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Boolean algebra (structure)
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Boolean algebra (structure)
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Boolean algebra (structure)
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Boolean algebra (structure)
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Boolean algebra (structure)
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René Descartes
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Mind–body dualism


René Descartes
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René Descartes
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René Descartes
Philosophical work
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René Descartes
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René Descartes
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René Descartes
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Emancipation from Church doctrine
Mathematical legacy
Contemporary reception
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René Descartes
Historical impact
Emancipation from Church doctrine


René Descartes
Historical impact
Mathematical legacy


René Descartes
Historical impact
Mathematical legacy
Influence on Newton's mathematics


René Descartes
Historical impact
Contemporary reception


René Descartes
Historical impact
Purported Rosicrucianism


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René Descartes
Bibliography
Collected editions


René Descartes
Bibliography
Early editions of specific works


René Descartes
Bibliography
Collected English translations


René Descartes
Bibliography
Translation of single works


Orders on the Cartesian product of totally ordered sets


Total order
Strict and non-strict total orders


Lattice theory
Finite total orders
Category theory
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Completeness
Sums of orders
Decidability


Total order
Further concepts
Lattice theory


Total order
Further concepts
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Total order
Further concepts
Category theory


Total order
Further concepts
Order topology


Total order
Further concepts
Completeness


Total order
Further concepts
Sums of orders


Total order
Further concepts
Decidability


Proof of irrationality for non-perfect nth power x


nth root
Definition and notation
Square roots


nth root
Definition and notation
Cube roots


nth root
Simplified form of a radical expression


Digit-by-digit calculation of principal roots of decimal (base 10) numbers


nth root
Computing principal roots
Using Newton's method


nth root
Computing principal roots
Logarithmic calculation


Sketch of a proof that e is transcendental


Transcendental number
Numbers proven to be transcendental


Transcendental number
Possible transcendental numbers


Existence of an algebraic closure and splitting fields


Etymology
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Polynomial
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Rational functions
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Polynomial
Related concepts
Rational functions


Polynomial
Related concepts
Laurent polynomials


Polynomial
Related concepts
Power series


Positional notation
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Polynomial
Applications
Positional notation


Polynomial
Applications
Interpolation and approximation


Polynomial
Applications
Other applications


Polynomial
History
History of the notation


Notation
Integer part definition
Quotient of two integers
More general quotients


Relation to the axiom schema of replacement


Axiom schema of specification
Unrestricted comprehension


Axiom schema of specification
In NBG class theory


Axiom schema of specification
In higher-order settings


Axiom schema of specification
In Quine's New Foundations


Axiom schema of specification
References


Axiom schema of specification
References
Citations


Introduction
Quine's classification
Ramsey's classification
In philosophy
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Overview
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Universe of discourse
Boole’s 1854 definition


Domain of discourse
Universe of discourse


Domain of discourse
Boole’s 1854 definition


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David Hilbert
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Internal set theory
Intuitive justification


Internal set theory
Intuitive justification
Principles of the standard predicate


Internal set theory
Formal axioms for IST

I: Idealisation
S: Standardisation
T: Transfer


Internal set theory
Formal axioms for IST
I: Idealisation

Applied to the relation ≠
Applied to the relation <
Applied to the  relation ∈


Internal set theory
Formal axioms for IST
I: Idealisation
Applied to the relation ≠


Internal set theory
Formal axioms for IST
I: Idealisation
Applied to the relation <


Internal set theory
Formal axioms for IST
I: Idealisation
Applied to the  relation ∈


Internal set theory
Formal axioms for IST
S: Standardisation


Internal set theory
Formal axioms for IST
T: Transfer


Internal set theory
Formal justification for the axioms


Technology
Number of users and publications


Semantic Scholar
Number of users and publications


Partial order relations
Notation
Alternative definitions
Examples
Derived notions
Mappings between partially ordered sets
Number of partial orders
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In category theory
Partial orders in topological spaces
Intervals


Partially ordered set
Partial order relations

Correspondence of strict and non-strict partial order relations


Partially ordered set
Partial order relations
Partial orders


Partially ordered set
Partial order relations
Strict partial orders


Correspondence of strict and non-strict partial order relations


Partially ordered set
Partial order relations
Dual orders


Partially ordered set
Alternative definitions


Orders on the Cartesian product of partially ordered sets


Partially ordered set
Examples
Sums of partially ordered sets


Partially ordered set
Derived notions
Extrema


Partially ordered set
Mappings between partially ordered sets


Partially ordered set
Number of partial orders


Partially ordered set
In category theory


Partially ordered set
Partial orders in topological spaces


Reviews
Online database
Mathematical citation quotient
Current Mathematical Publications


Mathematical Reviews
Mathematical citation quotient


Mathematical Reviews
Current Mathematical Publications


In natural language
Formal proof
Discussion
Resolution


Curry's paradox
In natural language
Informal proof


Sentential logic
Naive set theory
Lambda calculus
Combinatory logic


Curry's paradox
Formal proof
Sentential logic


Curry's paradox
Formal proof
Naive set theory


Curry's paradox
Formal proof
Lambda calculus


Curry's paradox
Formal proof
Combinatory logic


Curry's paradox
Discussion
Consequences for some formal logic


No resolution in lambda calculus
Resolution in unrestricted languages
Resolution in formal logic
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Curry's paradox
Resolution
No resolution in lambda calculus


Curry's paradox
Resolution
No resolution in lambda calculus
Domain of lambda calculus terms


Curry's paradox
Resolution
Resolution in unrestricted languages


Curry's paradox
Resolution
Resolution in formal logic


Curry's paradox
Resolution
Resolution in set theory


Definition
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Properties
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Lebesgue measure
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Lebesgue measure
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Zero of a function
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Life
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Sense and reference
1924 diary
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Gottlob Frege
Life
Studies at University (1869–74)


Logic, foundation of arithmetic
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Forms of fractions
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Forms of fractions
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Forms of fractions
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Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


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Arithmetic with fractions
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Arithmetic with fractions
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Arithmetic with fractions
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Arithmetic with fractions
Addition
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Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication

Multiplying a fraction by another fraction

Multiplying a fraction by a whole number


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
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Converting between decimals and fractions

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Fraction
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Pedagogical tools
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Fraction
In formal education
Pedagogical tools


Fraction
In formal education
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Decimal expansion and recurrence sequence

Every rational number is either a terminating or repeating decimal

Every repeating or terminating decimal is a rational number


Decimal expansion and recurrence sequence


Every rational number is either a terminating or repeating decimal


Every repeating or terminating decimal is a rational number


Fraction
Forms of fractions
Simple, common, or vulgar fractions


Fraction
Forms of fractions
Proper and improper fractions


Reciprocals and the "invisible denominator"


Fraction
Forms of fractions
Decimal fractions and percentages


Fraction
Forms of fractions
Mixed numbers


Fraction
Forms of fractions
Historical notions


Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


Fraction
Arithmetic with fractions
Equivalent fractions


Fraction
Arithmetic with fractions
Equivalent fractions
Simplifying (reducing) fractions


Fraction
Arithmetic with fractions
Comparing fractions


Fraction
Arithmetic with fractions
Addition


Fraction
Arithmetic with fractions
Addition
Adding unlike quantities


Fraction
Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
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Converting between decimals and fractions


Converting repeating decimals to fractions


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In formal education
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Forms of fractions
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Reciprocals and the "invisible denominator"


Fraction
Forms of fractions
Decimal fractions and percentages


Fraction
Forms of fractions
Mixed numbers


Fraction
Forms of fractions
Historical notions


Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


Fraction
Arithmetic with fractions
Equivalent fractions


Fraction
Arithmetic with fractions
Equivalent fractions
Simplifying (reducing) fractions


Fraction
Arithmetic with fractions
Comparing fractions


Fraction
Arithmetic with fractions
Addition


Fraction
Arithmetic with fractions
Addition
Adding unlike quantities


Fraction
Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
Division


Converting between decimals and fractions


Converting repeating decimals to fractions


Fraction
Fractions in abstract mathematics


Fraction
In formal education
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Fraction
In formal education
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Other systems


2018 Pacific hurricane season
Systems
Tropical Depression One-E


2018 Pacific hurricane season
Systems
Hurricane Aletta


2018 Pacific hurricane season
Systems
Hurricane Bud


2018 Pacific hurricane season
Systems
Tropical Storm Carlotta


2018 Pacific hurricane season
Systems
Tropical Storm Daniel


2018 Pacific hurricane season
Systems
Tropical Storm Emilia


2018 Pacific hurricane season
Systems
Hurricane Fabio


2018 Pacific hurricane season
Systems
Tropical Storm Gilma


2018 Pacific hurricane season
Systems
Tropical Depression Nine-E


2018 Pacific hurricane season
Systems
Hurricane Hector


2018 Pacific hurricane season
Systems
Tropical Storm Ileana


2018 Pacific hurricane season
Systems
Hurricane John


2018 Pacific hurricane season
Systems
Tropical Storm Kristy


2018 Pacific hurricane season
Systems
Hurricane Lane


2018 Pacific hurricane season
Systems
Hurricane Miriam


2018 Pacific hurricane season
Systems
Hurricane Norman


2018 Pacific hurricane season
Systems
Hurricane Olivia


2018 Pacific hurricane season
Systems
Tropical Storm Paul


2018 Pacific hurricane season
Systems
Tropical Depression Nineteen-E


2018 Pacific hurricane season
Systems
Hurricane Rosa


2018 Pacific hurricane season
Systems
Hurricane Sergio


2018 Pacific hurricane season
Systems
Hurricane Walaka


2018 Pacific hurricane season
Systems
Tropical Storm Tara


2018 Pacific hurricane season
Systems
Tropical Storm Vicente


2018 Pacific hurricane season
Systems
Hurricane Willa


2018 Pacific hurricane season
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Tropical Storm Xavier


2018 Pacific hurricane season
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Legal background and incident
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Background Destruction Attribution Consequences Reactions


Destruction of the Kakhovka Dam
Background


Destruction of the Kakhovka Dam
Destruction


Destruction of the Kakhovka Dam
Attribution


Destruction of the Kakhovka Dam
Attribution
Russia


Destruction of the Kakhovka Dam
Attribution
Ukraine


Destruction of the Kakhovka Dam
Consequences

Flooding and evacuations
Animals and environment
Counteroffensive
Water supply


Destruction of the Kakhovka Dam
Consequences
Flooding and evacuations


Destruction of the Kakhovka Dam
Consequences
Animals and environment


Destruction of the Kakhovka Dam
Consequences
Counteroffensive


Destruction of the Kakhovka Dam
Consequences
Water supply


Destruction of the Kakhovka Dam
Reactions


Destruction of the Kakhovka Dam
Reactions
Russia


Destruction of the Kakhovka Dam
Reactions
Ukraine


Destruction of the Kakhovka Dam
Reactions
International community


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Governor (India)
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Governor (India)
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Iraq and weapons of mass destruction
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Iraq and weapons of mass destruction
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Program development 1960s–1980s

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United States
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Iraq and weapons of mass destruction
History
Program development 1960s–1980s
Western help with Iraq's WMD program


Iraq and weapons of mass destruction
History
Program development 1960s–1980s
United States


Iraq and weapons of mass destruction
History
Program development 1960s–1980s
United Kingdom


Iraq and weapons of mass destruction
History
Iran–Iraq War


Iraq and weapons of mass destruction
History
Iran–Iraq War
Chemical weapon attacks


Iraq and weapons of mass destruction
History
1991 Gulf War


Iraq and weapons of mass destruction
History
UN inspections

UNSCOM inspections 1991–1998
Between inspections: 1998–2003


Iraq and weapons of mass destruction
History
UN inspections
UNSCOM inspections 1991–1998


Iraq and weapons of mass destruction
History
UN inspections
Between inspections: 1998–2003


Iraq and weapons of mass destruction
History
2003 Iraq War

2006: House Armed Services Committee Hearing


Iraq and weapons of mass destruction
History
2003 Iraq War
Prelude


Iraq and weapons of mass destruction
History
2003 Iraq War
Legal justification


Iraq and weapons of mass destruction
History
2003 Iraq War
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Iraq and weapons of mass destruction
History
2003 Iraq War
Iraq Survey Group


Iraq and weapons of mass destruction
History
2003 Iraq War
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Iraq and weapons of mass destruction
History
2003 Iraq War
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Iraq and weapons of mass destruction
History
2003 Iraq War
2005: Operation Avarice


2006: House Armed Services Committee Hearing


Iraq and weapons of mass destruction
History
2003 Iraq War
The New York Times investigative report


Iraq and weapons of mass destruction
History
2009 Declaration


Iraq and weapons of mass destruction
Public perception


History
Demographics
Culture
Native gastronomy
Tourism
Geography
Ecology
Economy
Education
Infrastructure
Major communities


Iraq's nuclear program
Strategy and diplomacy
Attack
International political reactions
Assessment


Iranian attack
Operational planning
Preliminary Israeli/Iranian actions


Operation Opera
Strategy and diplomacy
Iranian attack


Operation Opera
Strategy and diplomacy
Operational planning


Operation Opera
Strategy and diplomacy
Preliminary Israeli/Iranian actions


Operation Opera
International political reactions


Operation Opera
International political reactions
Aftermath


History
Geography
Demographics
Government and politics
Economy
Culture/Cityscape
Transport
Education
Sports and entertainment
Notable people


Balasore
Government and politics
Civic Utility / Amenities / Services


Balasore
Government and politics
Civic Utility / Amenities / Services
Healthcare


Balasore
Culture/Cityscape
Places of interest


Technical colleges/institutions
Universities/colleges
Schools


Balasore
Education
Technical colleges/institutions


Permit Field
Reserved Police Line Ground
Zilla School Stadium
ITI Field
Entertainment


Balasore
Sports and entertainment
Permit Field


Balasore
Sports and entertainment
Reserved Police Line Ground


Balasore
Sports and entertainment
Zilla School Stadium


Balasore
Sports and entertainment
ITI Field


Balasore
Sports and entertainment
Entertainment


History
International markets
Design and features
Safety
Uses
Gallery


Early life
Manchester United career
Munich air disaster
Legacy
Notes, citations and sources
External links


Geoff Bent
Notes, citations and sources
Notes


Geoff Bent
Notes, citations and sources
Citations


Geoff Bent
Notes, citations and sources
Sources


Geoff Bent
Notes, citations and sources
Sources
Books


Geoff Bent
Notes, citations and sources
Sources
Newspapers


Geoff Bent
Notes, citations and sources
Sources
Online


Events
Births and deaths
Nobel Prizes
Fields Medal
References
External links


January
February
March
April
May
June
July
August
September
October
November
December
Date unknown


Geography
Demographics
Principal communities
Economy
Tourist attractions
Culture
Communications and transportation
History
Archaeology
Education
Sister cities


Colima
Communications and transportation


Colima
Communications and transportation
Transport


Colima
Communications and transportation
Media


Name and seal
Pre-Hispanic period
Colonial era
Independence to the present


Colima
History
Independence to the present


Etymology
History
Geography
Climate
Demographics
Economy
Education
Transportation
Architecture
Culture
References
Bibliography
External links


Museums
Botanical gardens
Clubs
Places of worship


Important presidential interventions in the past


Duty
Legislative powers
Executive powers
Judicial powers
Appointment powers
Financial powers
Diplomatic powers
Military powers
Pardoning powers
Emergency powers


President of India
Powers and duties
Duty


President of India
Powers and duties
Legislative powers


President of India
Powers and duties
Executive powers


President of India
Powers and duties
Judicial powers


President of India
Powers and duties
Appointment powers


President of India
Powers and duties
Financial powers


President of India
Powers and duties
Diplomatic powers


President of India
Powers and duties
Military powers


President of India
Powers and duties
Pardoning powers


President of India
Powers and duties
Emergency powers

National emergency
State emergency
Financial emergency


President of India
Powers and duties
Emergency powers
National emergency


President of India
Powers and duties
Emergency powers
State emergency


President of India
Powers and duties
Emergency powers
Financial emergency


President of India
Selection process
Eligibility


Conditions for the presidency
Election process
Oath or affirmation


President of India
Time of election
Conditions for the presidency


President of India
Time of election
Election process


President of India
Time of election
Oath or affirmation


President of India
President vs Chief Justice


President of India
President vs Prime minister


Important presidential interventions in the past

Proving majority in the parliament
Proof of Majority to form a Government
Vetoing of a Bill
Rashtrapati Bhavan Communiqués
Return of a Bill
Sacking state governors


History
Ideologies and movements
Academic use of the term


United States
British Commonwealth
Germany
Australia and New Zealand


Early history
20th century
21st century
Patterns of influence


White supremacy
History
United States
Early history


White supremacy
History
United States
20th century


White supremacy
History
United States
21st century


White supremacy
History
United States
Patterns of influence


White supremacy
History
British Commonwealth


White supremacy
History
British Commonwealth
South Africa


White supremacy
History
British Commonwealth
Rhodesia


White supremacy
History
Australia and New Zealand


White supremacy
Ideologies and movements


White supremacy
Ideologies and movements
White nationalism


White supremacy
Ideologies and movements
White separatism

Notable white separatists
Aligned organizations and philosophies


White supremacy
Ideologies and movements
White separatism
Notable white separatists


White supremacy
Ideologies and movements
White separatism
Aligned organizations and philosophies


White supremacy
Academic use of the term


Background Murder Perpetrators Reactions


Murder of James Byrd Jr.
Perpetrators
Shawn Berry


Murder of James Byrd Jr.
Perpetrators
Lawrence Brewer


Murder of James Byrd Jr.
Perpetrators
John King


Impact on US politics
Musical and poetry tributes


Murder of James Byrd Jr.
Reactions
Impact on US politics


Murder of James Byrd Jr.
Reactions
Musical and poetry tributes


History
Geography
Climate
Demographics
Economy
Government
Parks and recreation
Education
Media
Notable people
References
External links


Franchise history
Team history
Team identity
Home ground
Kit manufacturers and sponsors
Seasons
Current squad
Administration and support staff
Statistics
References
External links


Gujarat Titans
Team history
2022 IPL season


Gujarat Titans
Kit manufacturers and sponsors


Gujarat Titans
Seasons
Indian Premier League


Gujarat Titans
Administration and support staff


Overall results in the IPL
By opposition


Gujarat Titans
Statistics
Overall results in the IPL


Parametric representations
Properties
In more than three dimensions
Hyperboloid structures
Relation to the sphere


Hyperboloid
Parametric representations
Generalised equations


Hyperboloid of one sheet
Hyperboloid of two sheets
Other properties


Hyperboloid
Properties
Hyperboloid of one sheet


Hyperboloid
Properties
Hyperboloid of one sheet
Lines on the surface


Hyperboloid
Properties
Hyperboloid of one sheet
Plane sections


Hyperboloid
Properties
Hyperboloid of two sheets


Hyperboloid
Properties
Other properties
Symmetries


Hyperboloid
Properties
Other properties
Curvature


Hyperboloid
In more than three dimensions


Motivation History Concepts Topics Applications


Topologies on sets
Continuous functions and homeomorphisms
Manifolds


Topology
Concepts
Continuous functions and homeomorphisms


General topology
Algebraic topology
Differential topology
Geometric topology
Generalizations


Biology
Computer science
Physics
Robotics
Games and puzzles
Fiber art


Scalar (mathematics)
Definitions and properties

Scalars of vector spaces
Scalars as vector components
Scalars in normed vector spaces
Scalars in modules
Scaling transformation


Scalar (mathematics)
Definitions and properties
Scalars of vector spaces


Scalar (mathematics)
Definitions and properties
Scalars as vector components


Scalar (mathematics)
Definitions and properties
Scalars in normed vector spaces


Scalar (mathematics)
Definitions and properties
Scalars in modules


Scalar (mathematics)
Definitions and properties
Scaling transformation


History
Vector spaces
Matrices
Linear systems
Endomorphisms and square matrices
Duality
Relationship with geometry
Usage and applications
Extensions and generalizations


Linear algebra
Vector spaces
Linear maps


Linear algebra
Vector spaces
Subspaces, span, and basis


Linear algebra
Endomorphisms and square matrices

Determinant
Eigenvalues and eigenvectors


Linear algebra
Endomorphisms and square matrices
Determinant


Linear algebra
Endomorphisms and square matrices
Eigenvalues and eigenvectors


Linear algebra
Duality
Inner-product spaces


Linear algebra
Relationship with geometry


Geometry of ambient space
Functional analysis
Study of complex systems
Scientific computation


Linear algebra
Usage and applications
Geometry of ambient space


Linear algebra
Usage and applications
Functional analysis


Linear algebra
Usage and applications
Study of complex systems


Linear algebra
Usage and applications
Scientific computation


Linear algebra
Extensions and generalizations

Module theory
Multilinear algebra and tensors
Topological vector spaces


Linear algebra
Extensions and generalizations
Module theory


Linear algebra
Extensions and generalizations
Multilinear algebra and tensors


Linear algebra
Extensions and generalizations
Topological vector spaces


Types
Properties
Cylindrical surfaces
Projective geometry
Prisms


Right circular hollow cylinder (cylindrical shell)


Cylinder
Properties
On the Sphere and Cylinder


Elliptic cylinder
Hyperbolic cylinder
Parabolic cylinder


Cylinder
Cylindrical surfaces
Elliptic cylinder


Cylinder
Cylindrical surfaces
Hyperbolic cylinder


Cylinder
Cylindrical surfaces
Parabolic cylinder


Definitions
Properties
Enumeration
Types of trees


Tree
Forest
Polytree
Polyforest
Rooted tree
Ordered tree


Tree (graph theory)
Definitions
Polytree


Tree (graph theory)
Definitions
Polyforest


Tree (graph theory)
Definitions
Rooted tree


Tree (graph theory)
Definitions
Ordered tree


Tree (graph theory)
Enumeration
Labeled trees


Tree (graph theory)
Enumeration
Unlabeled trees


Topological spaces with algebraic structure


Definition via neighbourhoods
Definition via open sets
Definition via closed sets
Other definitions


Topological space
Definitions
Definition via neighbourhoods


Topological space
Definitions
Definition via open sets


Topological space
Definitions
Definition via open sets
Examples of topologies


Topological space
Definitions
Definition via closed sets


Topological space
Definitions
Other definitions


Topological space
Comparison of topologies


Topological space
Examples of topological spaces

Metric spaces
Proximity spaces
Uniform spaces
Function spaces
Cauchy spaces
Convergence spaces
Grothendieck sites
Other spaces


Topological space
Examples of topological spaces
Metric spaces


Topological space
Examples of topological spaces
Proximity spaces


Topological space
Examples of topological spaces
Uniform spaces


Topological space
Examples of topological spaces
Function spaces


Topological space
Examples of topological spaces
Cauchy spaces


Topological space
Examples of topological spaces
Convergence spaces


Topological space
Examples of topological spaces
Grothendieck sites


Topological space
Examples of topological spaces
Other spaces


Topological space
Topological constructions


Topological space
Classification of topological spaces


Topological spaces with algebraic structure


Topological space
Topological spaces with order structure


Symbols A B C D E F G H I K L M N O P Q R S T U V W


Motivating examples
Mathematical definition
Charts, atlases, and transition maps
Manifold with boundary
Construction
History
Additional structure
Classification and invariants
Surfaces
Maps of manifolds
Generalizations of manifolds


Manifold
Motivating examples
Other curves


Manifold
Charts, atlases, and transition maps

Charts
Atlases
Transition maps
Additional structure


Manifold
Charts, atlases, and transition maps
Charts


Manifold
Charts, atlases, and transition maps
Atlases


Manifold
Charts, atlases, and transition maps
Transition maps


Manifold
Charts, atlases, and transition maps
Additional structure


Manifold
Manifold with boundary
Boundary and interior


Charts
Patchwork
Identifying points of a manifold
Gluing along boundaries
Cartesian products


Manifold
Construction
Charts
Sphere with charts


Intrinsic and extrinsic view
n-Sphere as a patchwork


Manifold
Construction
Patchwork
Intrinsic and extrinsic view


Manifold
Construction
Patchwork
n-Sphere as a patchwork


Manifold
Construction
Identifying points of a manifold


Manifold
Construction
Gluing along boundaries


Manifold
Construction
Cartesian products


Early development
Synthesis
Poincaré's definition
Topology of manifolds: highlights


Manifold
History
Topology of manifolds: highlights


Topological manifolds
Differentiable manifolds
Riemannian manifolds
Finsler manifolds
Lie groups
Other types of manifolds


Manifold
Additional structure
Topological manifolds


Manifold
Additional structure
Differentiable manifolds


Manifold
Additional structure
Riemannian manifolds


Manifold
Additional structure
Finsler manifolds


Manifold
Additional structure
Lie groups


Manifold
Additional structure
Other types of manifolds


Orientability
Genus and the Euler characteristic


Möbius strip
Klein bottle
Real projective plane


Manifold
Surfaces
Orientability
Möbius strip


Manifold
Surfaces
Orientability
Klein bottle


Manifold
Surfaces
Orientability
Real projective plane


Manifold
Surfaces
Genus and the Euler characteristic


Manifold
Maps of manifolds
Scalar-valued functions


Vectors in Euclidean geometry
Vector spaces
Vectors in algebra
Data represented by vectors


Vector (mathematics and physics)
Vectors in Euclidean geometry


Vector (mathematics and physics)
Vector spaces


Vector (mathematics and physics)
Vectors in algebra


Vector (mathematics and physics)
Data represented by vectors


Definitions Applications History Representation Problems


Computer science
Linguistics
Physics and chemistry
Social sciences
Biology
Mathematics
Other topics


Graph theory
Applications
Computer science


Graph theory
Applications
Physics and chemistry


Graph theory
Applications
Social sciences


Visual: Graph drawing
Tabular: Graph data structures


Graph theory
Representation
Visual: Graph drawing


Graph theory
Representation
Tabular: Graph data structures


Subgraphs, induced subgraphs, and minors


Graph theory
Problems
Subsumption and unification


Graph theory
Problems
Visibility problems


Graph theory
Problems
Decomposition problems


Representation
Conversion
Elementary arithmetic
Real numbers
Cultural history
Base16 (transfer encoding)


Other symbols for 10–15 and mostly different symbol sets


Hexadecimal
Representation
Written representation


Hexadecimal
Representation
Written representation
Distinguishing from decimal


Hexadecimal
Representation
Verbal and digital representations


Hexadecimal
Representation
Hexadecimal exponential notation


Conversion through addition and multiplication


Hexadecimal
Conversion
Binary conversion


Hexadecimal
Conversion
Other simple conversions


Hexadecimal
Conversion
Division-remainder in source base


Hexadecimal
Conversion
Tools for conversion


Rational numbers
Irrational numbers
Powers


Hexadecimal
Real numbers
Rational numbers


Hexadecimal
Real numbers
Irrational numbers


Conversion to and from other numeral systems


Egypt
China
India
Other cultures
Western predecessors to Leibniz
Leibniz and the I Ching
Later developments


Binary number
History
Western predecessors to Leibniz


Binary number
History
Leibniz and the I Ching


Binary number
History
Later developments


Binary number
Counting in binary
Decimal counting


Binary number
Counting in binary
Binary counting


Addition
Subtraction
Multiplication
Division
Square root


Binary number
Binary arithmetic
Addition


Binary number
Binary arithmetic
Addition
Long carry method


Binary number
Binary arithmetic
Addition
Addition table


Binary number
Binary arithmetic
Subtraction


Binary number
Binary arithmetic
Multiplication


Binary number
Binary arithmetic
Multiplication
Multiplication table


Binary number
Binary arithmetic
Division


Binary number
Binary arithmetic
Square root


Conversion to and from other numeral systems

Decimal to Binary
Binary to Decimal
Hexadecimal
Octal


Administration
Background
Participating teams
Personnel changes
Rules
Venues
Format
Points table
Match summary
League stage
Playoffs
Broadcasting
Criticism
Statistics and awards
Notes
References
External links


2023 Indian Premier League
Administration


2023 Indian Premier League
Participating teams


2023 Indian Premier League
Personnel changes


2023 Indian Premier League
Match summary


Qualifier 1
Eliminator
Qualifier 2
Final


2023 Indian Premier League
Playoffs
Qualifier 1


2023 Indian Premier League
Playoffs
Eliminator


2023 Indian Premier League
Playoffs
Qualifier 2


2023 Indian Premier League
Playoffs
Final


2023 Indian Premier League
Criticism
Slow over- rate


2023 Indian Premier League
Criticism
Impact player rule


2023 Indian Premier League
Statistics and awards

Most runs
Most wickets
End of the season awards


2023 Indian Premier League
Statistics and awards
Most runs


2023 Indian Premier League
Statistics and awards
Most wickets


2023 Indian Premier League
Statistics and awards
End of the season awards


2023 Indian Premier League
External links


History
Home ground
Players
Identity
Kit and sponsors
Support and fan following
Brand value
Controversies
Result summary
Statistics
Rivalries
Seasons
Current squad
Administration and support staff
In popular culture


2008–2009: First seasons
2010: IPL and CLT20 double
2011–2015: Success and suspension
2018–present


Chennai Super Kings
History
2008–2009: First seasons


Chennai Super Kings
History
2010: IPL and CLT20 double


Chennai Super Kings
History
2011–2015: Success and suspension


Chennai Super Kings
History
2018–present


Chennai Super Kings
Identity
Name, logo, crest and colors


Chennai Super Kings
Support and fan following


Ownership of N. Srinivasan
2013 betting scandal
IPL suspension


Chennai Super Kings
Controversies
Ownership of N. Srinivasan


Chennai Super Kings
Controversies
2013 betting scandal


Chennai Super Kings
Controversies
IPL suspension


Overall results in the Champions League T20


Chennai Super Kings
Result summary
Overall results


Chennai Super Kings
Result summary
By opposition


Chennai Super Kings
Statistics
Most runs


Chennai Super Kings
Statistics
Most wickets


Rivalry with Royal Challengers Bangalore


Chennai Super Kings
Rivalries
Rivalry with Mumbai Indians


Indian Premier League
Champions League T20


Chennai Super Kings
Seasons
Indian Premier League


Chennai Super Kings
Seasons
Champions League T20


Chennai Super Kings
Administration and support staff


Background
Date
Electoral system
Candidates
Endorsements
Second round
Controversies
Opinion polls
Results
Aftermath


2023 Turkish presidential election
Background


2023 Turkish presidential election
Background
2018 elections


Discussion on snap elections
Discussion on postponing election date
Election calendar


2023 Turkish presidential election
Date
Discussion on snap elections


2023 Turkish presidential election
Date
Discussion on postponing election date


2023 Turkish presidential election
Date
Election calendar


2023 Turkish presidential election
Electoral system


2023 Turkish presidential election
Candidates


2023 Turkish presidential election
Candidates
Candidates


2023 Turkish presidential election
Candidates
Nominations


2023 Turkish presidential election
Endorsements


2023 Turkish presidential election
Endorsements
Parties and alliances


2023 Turkish presidential election
Second round

Negotiations by Oğan and Özdağ
Erdoğan's strategy
Kılıçdaroğlu's strategy
Potential debate


2023 Turkish presidential election
Second round
Negotiations by Oğan and Özdağ

Dissolution of ATA Alliance
Endorsement of Erdoğan by Oğan
Özdağ's support for Kılıçdaroğlu


2023 Turkish presidential election
Second round
Negotiations by Oğan and Özdağ
Dissolution of ATA Alliance


2023 Turkish presidential election
Second round
Negotiations by Oğan and Özdağ
Endorsement of Erdoğan by Oğan


2023 Turkish presidential election
Second round
Negotiations by Oğan and Özdağ
Özdağ's support for Kılıçdaroğlu


2023 Turkish presidential election
Second round
Erdoğan's strategy


2023 Turkish presidential election
Second round
Kılıçdaroğlu's strategy


2023 Turkish presidential election
Second round
Potential debate


2023 Turkish presidential election
Controversies

Azerbaijani and Russian interference allegations


2023 Turkish presidential election
Controversies
Erdoğan's right to seek re-election


2023 Turkish presidential election
Controversies
Political violence


2023 Turkish presidential election
Controversies
Erdoğan's photo on ballot paper


2023 Turkish presidential election
Controversies
Fake videos


2023 Turkish presidential election
Controversies
Block of Ekşi Sözlük


2023 Turkish presidential election
Controversies
Reporting of the results


Azerbaijani and Russian interference allegations


2023 Turkish presidential election
Opinion polls


2023 Turkish presidential election
Opinion polls
First round


2023 Turkish presidential election
Opinion polls
Second round


2023 Turkish presidential election
Results


2023 Turkish presidential election
Results
Analysis


2023 Turkish presidential election
Aftermath


2023 Turkish presidential election
Aftermath
Domestic


2023 Turkish presidential election
Aftermath
Domestic
International


FBI career and first espionage activities (1976–1981)

FBI counterintelligence unit, further espionage activities (1985–1991)

Later FBI career, continued espionage activities (1992–2001)


FBI career and first espionage activities (1976–1981)


FBI counterintelligence unit, further espionage activities (1985–1991)


Later FBI career, continued espionage activities (1992–2001)


Robert Hanssen
Guilty plea and imprisonment


Robert Hanssen
References and further reading


History
Leadership roles
Selection process
Incumbency
Duties and responsibilities
Acting President
Latest election
Timeline of presidents


President of Turkey
Leadership roles
Head of state


President of Turkey
Leadership roles
Head of party


President of Turkey
Selection process
Eligibility


President of Turkey
Selection process
Election


President of Turkey
Selection process
Inauguration


Term limit
Vacancies and successions
Accountability and non-accountability
Residence


President of Turkey
Incumbency
Term limit


President of Turkey
Incumbency
Vacancies and successions


President of Turkey
Incumbency
Accountability and non-accountability

Before the 2017 constitutional amendment


President of Turkey
Incumbency
Accountability and non-accountability
After the 2017 constitutional amendment


Before the 2017 constitutional amendment


President of Turkey
Incumbency
Residence


President of Turkey
Duties and responsibilities


After the 2017 constitutional referendum

Before the 2017 constitutional referendum


After the 2017 constitutional referendum


Before the 2017 constitutional referendum


President of Turkey
Timeline of presidents


Race background
Race schedules
Entry list
Testing and rookie orientation
Practice
Time trials
Post-qualifying practice
Starting grid
Race report
Box score
Broadcasting
References
External links


Sponsorship
Rule changes
2023 IndyCar Series


2023 Indianapolis 500
Race background
Sponsorship


2023 Indianapolis 500
Race background
Rule changes


2023 Indianapolis 500
Race background
2023 IndyCar Series


2023 Indianapolis 500
Testing and rookie orientation

Open Test—Thursday April 20
Open test—Friday April 21


2023 Indianapolis 500
Testing and rookie orientation
Open Test—Thursday April 20


2023 Indianapolis 500
Testing and rookie orientation
Open test—Friday April 21


Opening day—Tuesday May 16
Wednesday May 17
Thursday May 18
Fast Friday—Friday May 19


2023 Indianapolis 500
Practice
Opening day—Tuesday May 16


2023 Indianapolis 500
Practice
Wednesday May 17


2023 Indianapolis 500
Practice
Thursday May 18


2023 Indianapolis 500
Practice
Fast Friday—Friday May 19


2023 Indianapolis 500
Time trials
Saturday, May 20


2023 Indianapolis 500
Time trials
Sunday, May 21

Top 12 qualifying
Last Chance Qualifying
Fast Six Qualifying


2023 Indianapolis 500
Time trials
Sunday, May 21
Top 12 qualifying


2023 Indianapolis 500
Time trials
Sunday, May 21
Last Chance Qualifying


2023 Indianapolis 500
Time trials
Sunday, May 21
Fast Six Qualifying


2023 Indianapolis 500
Post-qualifying practice

Post-qualifying practice—Monday, May 22
Special test—Thursday, May 25
Carb Day—Friday, May 26
Pit Stop Challenge


2023 Indianapolis 500
Post-qualifying practice
Post-qualifying practice—Monday, May 22


2023 Indianapolis 500
Post-qualifying practice
Special test—Thursday, May 25


2023 Indianapolis 500
Post-qualifying practice
Carb Day—Friday, May 26


2023 Indianapolis 500
Post-qualifying practice
Pit Stop Challenge


2023 Indianapolis 500
Race report
First half


2023 Indianapolis 500
Race report
Second half

First red flag
Second red flag
Third red flag and finish


2023 Indianapolis 500
Race report
Second half
First red flag


2023 Indianapolis 500
Race report
Second half
Second red flag


2023 Indianapolis 500
Race report
Second half
Third red flag and finish


2023 Indianapolis 500
Broadcasting
Television


2023 Indianapolis 500
Broadcasting
Radio


Early life and historical context
Plessy v. Ferguson
Later life
Legacy
References
Further reading


Homer Plessy
Early life and historical context


Orchestrating a test case
Trial
Supreme Court appeal


Homer Plessy
Plessy v. Ferguson
Orchestrating a test case


Homer Plessy
Plessy v. Ferguson
Supreme Court appeal


Homer Plessy
References
Sources
Secondary sources


Homer Plessy
References
Sources
Tertiary sources


Songs written by Barry Mann and Cynthia Weil


Early life
Life as an empress
Literary work
Legacy


Marriage
Children
Pilgrimage to Jerusalem (438–439)
Banishment
In Jerusalem (443–460)


Aelia Eudocia
Life as an empress
Marriage

Blending Christianity with classical culture


Blending Christianity with classical culture


Aelia Eudocia
Life as an empress
Children


Aelia Eudocia
Life as an empress
Pilgrimage to Jerusalem (438–439)


Aelia Eudocia
Life as an empress
Banishment


Aelia Eudocia
Life as an empress
In Jerusalem (443–460)


Martyrdom of St. Cyprian
The Hamat Gader poem
Homeric centos


Aelia Eudocia
Literary work
Martyrdom of St. Cyprian


Aelia Eudocia
Literary work
The Hamat Gader poem


Aelia Eudocia
Literary work
Homeric centos


Wars with the Huns, Vandals, and Persians


Theodosius II
Reign
University and Law Code


Theodosius II
Reign
Famous "apple incident"


Theodosius II
Reign
Theological disputes


Early and family life
Career
Family
Notes
References
External links


Thomas West, 3rd Baron De La Warr
Early and family life


Thomas West, 3rd Baron De La Warr
Career


Thomas West, 3rd Baron De La Warr
Family


Thomas West, 3rd Baron De La Warr
References


Thomas West, 3rd Baron De La Warr
External links


Events
Births
Deaths
Holidays and observances
References
External links


Early life and education
American Museum of Natural History
Origami
Later life
References
External links


Alice Gray
American Museum of Natural History


Usage
Development
Ceremony
Other countries
Notes
References
Further reading
External links


United Kingdom
Other Commonwealth realms


Historical development
Relationship to royal consent


Royal assent
Usage
United Kingdom
Historical development


Royal assent
Usage
United Kingdom
Relationship to royal consent


Royal assent
Usage
Other Commonwealth realms


Royal assent
Usage
Other Commonwealth realms
Canada


United Kingdom
Australia and New Zealand
Canada


Royal assent
Ceremony
Australia and New Zealand


Belgium Japan Jordan Luxembourg Norway Spain Tonga


January–March
April–June
July–September
October–December
Date unknown


January
February
March
April
May
June
July
August
September
October
November
December
Date unknown


Early years
Career
Honors
Bibliography
Notes
Further reading
External links


Knud Rasmussen
Career
The Thule expeditions


Unreformed House of Commons
Movement for reform
Passage of the Reform Act
Results
Assessment


Reform Act 1832
Unreformed House of Commons

Composition
The franchise
Pocket boroughs, bribery


Reform Act 1832
Unreformed House of Commons
Composition


Reform Act 1832
Unreformed House of Commons
Composition
Counties


Reform Act 1832
Unreformed House of Commons
Composition
Boroughs


Reform Act 1832
Unreformed House of Commons
The franchise


Reform Act 1832
Unreformed House of Commons
The franchise
Women's suffrage


Reform Act 1832
Unreformed House of Commons
Pocket boroughs, bribery


Early attempts at reform
After the French Revolution
Reform during the 1820s


Reform Act 1832
Movement for reform
Early attempts at reform


Reform Act 1832
Movement for reform
After the French Revolution


Reform Act 1832
Movement for reform
Reform during the 1820s


Reform Act 1832
Passage of the Reform Act

First Reform Bill
Second Reform Bill
Third Reform Bill


Reform Act 1832
Passage of the Reform Act
First Reform Bill


Reform Act 1832
Passage of the Reform Act
Second Reform Bill


Reform Act 1832
Passage of the Reform Act
Third Reform Bill


Provisions
Effects
Tenant voters
Limitations
Further reform


Abolition of seats
Creation of new seats
Extension of the franchise


Reform Act 1832
Results
Provisions
Abolition of seats


Reform Act 1832
Results
Provisions
Creation of new seats


Reform Act 1832
Results
Provisions
Extension of the franchise


Beliefs
Practices
Sanctuaries and temples
Role of women
Mystery religions
History


Theology
Afterlife
Mythology
Morality
Sacred texts


Ancient Greek religion
Beliefs
Afterlife


Ancient Greek religion
Beliefs
Mythology


Ancient Greek religion
Beliefs
Sacred texts


Ceremonies
Sacrifice
Festivals
Rites of passage


Ancient Greek religion
Practices
Ceremonies


Ancient Greek religion
Practices
Sacrifice


Ancient Greek religion
Practices
Festivals


Ancient Greek religion
Practices
Rites of passage


Ancient Greek religion
Sanctuaries and temples


Ancient Greek religion
Sanctuaries and temples
Cult images


Ancient Greek religion
Mystery religions


Origins
Archaic and classical periods
Hellenistic period
Roman Empire
Suppression and decline
Modern revivals


Ancient Greek religion
History
Archaic and classical periods


Ancient Greek religion
History
Hellenistic period


Ancient Greek religion
History
Roman Empire


Ancient Greek religion
History
Suppression and decline


Ancient Greek religion
History
Modern revivals


History
Decimal value
Proofs of irrationality
Multiplicative inverse
Properties
Representations
Applications


Square root of 2
History
Ancient Roman architecture


Computation algorithms
Rational approximations
Records in computation


Square root of 2
Decimal value
Computation algorithms


Square root of 2
Decimal value
Rational approximations


Square root of 2
Decimal value
Records in computation


Square root of 2
Proofs of irrationality

Proof by infinite descent
Proof by unique factorization
Geometric proof
Constructive proof
Proof by Pythagorean triples


Square root of 2
Proofs of irrationality
Proof by infinite descent


Square root of 2
Proofs of irrationality
Proof by unique factorization


Square root of 2
Proofs of irrationality
Geometric proof


Square root of 2
Proofs of irrationality
Constructive proof


Square root of 2
Proofs of irrationality
Proof by Pythagorean triples


Series and product
Continued fraction
Nested square


Square root of 2
Representations
Series and product


Square root of 2
Representations
Continued fraction


Square root of 2
Representations
Nested square


Square root of 2
Applications
Paper size


Square root of 2
Applications
Physical sciences


Decimal expansion and recurrence sequence

Every rational number is either a terminating or repeating decimal

Every repeating or terminating decimal is a rational number


Decimal expansion and recurrence sequence


Every rational number is either a terminating or repeating decimal


Every repeating or terminating decimal is a rational number


Calculation
History
Mathematics
Applications and observations
Disputed observations


Relationship to Fibonacci and Lucas numbers


Contradiction from an expression in lowest terms


Golden ratio
Mathematics
Irrationality
By irrationality of √5


Golden ratio
Mathematics
Minimal polynomial


Golden ratio
Mathematics
Golden ratio conjugate and powers


Golden ratio
Mathematics
Continued fraction and square root


Construction
Golden angle
Pentagonal symmetry system
In triangles and quadrilaterals
Golden spiral
In the dodecahedron and icosahedron


Golden ratio
Mathematics
Geometry
Construction


Golden ratio
Mathematics
Geometry
Golden angle


Golden ratio
Mathematics
Geometry
Pentagonal symmetry system


Golden ratio
Mathematics
Geometry
In triangles and quadrilaterals


Golden ratio
Mathematics
Geometry
Golden spiral


Golden ratio
Mathematics
Geometry
In the dodecahedron and icosahedron


Golden ratio
Mathematics
Other properties


Golden ratio
Applications and observations

Architecture
Art
Books and design
Flags
Music
Nature
Physics
Optimization


Golden ratio
Applications and observations
Architecture


Golden ratio
Applications and observations
Art


Golden ratio
Applications and observations
Books and design


Golden ratio
Applications and observations
Flags


Golden ratio
Applications and observations
Music


Golden ratio
Applications and observations
Nature


Golden ratio
Applications and observations
Physics


Golden ratio
Applications and observations
Optimization


Egyptian pyramids
The Parthenon
Modern art


Golden ratio
Disputed observations
Egyptian pyramids


Golden ratio
Disputed observations
The Parthenon


Golden ratio
Disputed observations
Modern art


Multiplication of different kinds of numbers


Product of two natural numbers
Product of two integers
Product of two fractions
Product of two real numbers
Product of two complex numbers
Product of two quaternions


Multiplication
Definitions
Product of two natural numbers


Multiplication
Definitions
Product of two integers


Multiplication
Definitions
Product of two fractions


Multiplication
Definitions
Product of two real numbers


Multiplication
Definitions
Product of two complex numbers


Multiplication
Definitions
Product of two quaternions


Historical algorithms
Modern methods
Computer algorithms


Multiplication
Computation
Historical algorithms


Multiplication
Computation
Historical algorithms
Egyptians


Multiplication
Computation
Historical algorithms
Babylonians


Multiplication
Computation
Historical algorithms
Chinese


Multiplication
Computation
Modern methods


Multiplication
Computation
Modern methods
Grid method


Multiplication
Computation
Computer algorithms


Multiplication
Product of a sequence
Capital pi notation


Multiplication
Product of a sequence
Capital pi notation
Properties of capital pi notation


Multiplication
Product of a sequence
Infinite products


Multiplication
Multiplication with set theory


Multiplication
Multiplication in group theory


Multiplication of different kinds of numbers


Subfield
Extension field
Caveats
Examples
Algebraic extension
Transcendental extension
Normal, separable and Galois extensions
Generalizations
Extension of scalars


Field extension
Transcendental extension


Field extension
Normal, separable and Galois extensions


January–March
April–June
July–September
October–December
Date unknown


Philosophy of mathematics
Descriptive informalities
Proof terminology
Proof techniques


List of mathematical jargon
Philosophy of mathematics


List of mathematical jargon
Descriptive informalities


List of mathematical jargon
Proof terminology


List of mathematical jargon
Proof techniques


Examples
Definition and notation
Properties
Graphical representation
Invariants
Quotient space in topology


Equivalence class
Definition and notation


Equivalence class
Graphical representation


Equivalence class
Quotient space in topology


Notation and terminology
Interpretations
Properties
Performing addition
Addition of numbers
Generalizations
Related operations
In music


Addition
Interpretations
Extending a length


Commutativity
Associativity
Identity element
Successor
Units


Innate ability
Childhood learning
Decimal system
Non-decimal
Computers


Addition
Performing addition
Innate ability


Addition
Performing addition
Childhood learning


Addition
Performing addition
Childhood learning
Table


Addition
Performing addition
Decimal system

Carry
Decimal fractions
Scientific notation


Addition
Performing addition
Decimal system
Carry


Addition
Performing addition
Decimal system
Decimal fractions


Addition
Performing addition
Decimal system
Scientific notation


Addition
Performing addition
Non-decimal


Natural numbers
Integers
Rational numbers (fractions)
Real numbers
Complex numbers


Addition
Addition of numbers
Natural numbers


Addition
Addition of numbers
Rational numbers (fractions)


Addition
Addition of numbers
Real numbers


Addition
Addition of numbers
Complex numbers


Abstract algebra
Set theory and category theory


Addition
Generalizations
Abstract algebra

Vectors
Matrices
Modular arithmetic
General theory


Addition
Generalizations
Abstract algebra
Vectors


Addition
Generalizations
Abstract algebra
Matrices


Addition
Generalizations
Abstract algebra
Modular arithmetic


Addition
Generalizations
Abstract algebra
General theory


Addition
Generalizations
Set theory and category theory


Addition
Related operations
Other ways to add


In real numbers
In a metric space
Completeness
Generalizations


Cauchy sequence
In real numbers
Modulus of Cauchy convergence


Examples
Non-example: rational numbers
Non-example: open interval
Other properties


Cauchy sequence
Completeness
Non-example: rational numbers


Cauchy sequence
Completeness
Non-example: open interval


Cauchy sequence
Completeness
Other properties


In topological vector spaces
In topological groups
In groups
In a hyperreal continuum
Cauchy completion of categories


Cauchy sequence
Generalizations
In topological vector spaces


Cauchy sequence
Generalizations
In topological groups


Cauchy sequence
Generalizations
In groups


Cauchy sequence
Generalizations
In a hyperreal continuum


Cauchy sequence
Generalizations
Cauchy completion of categories


History
Important concepts
Main branches
Other topics
Applications
Famous Textbooks


Mathematical analysis
History
Modern
Foundations


Mathematical analysis
History
Modern
Modernization


Mathematical analysis
Important concepts


Mathematical analysis
Important concepts
Metric spaces


Mathematical analysis
Important concepts
Sequences and limits


Real analysis
Complex analysis
Functional analysis
Harmonic analysis
Differential equations
Measure theory
Numerical analysis
Vector analysis
Scalar analysis
Tensor analysis


Mathematical analysis
Main branches
Real analysis


Mathematical analysis
Main branches
Complex analysis


Mathematical analysis
Main branches
Functional analysis


Mathematical analysis
Main branches
Harmonic analysis


Mathematical analysis
Main branches
Differential equations


Mathematical analysis
Main branches
Measure theory


Mathematical analysis
Main branches
Numerical analysis


Mathematical analysis
Main branches
Vector analysis


Mathematical analysis
Main branches
Scalar analysis


Mathematical analysis
Main branches
Tensor analysis


Physical sciences
Signal processing
Other areas of mathematics


Mathematical analysis
Applications
Physical sciences


Mathematical analysis
Applications
Signal processing


Mathematical analysis
Applications
Other areas of mathematics


Definition
Representations
Ordering of cuts
Construction of the real numbers
Relation to interval arithmetic
Generalizations
Notes
References
External links


Dedekind cut
Construction of the real numbers


Dedekind cut
Relation to interval arithmetic


Arbitrary linearly ordered sets
Surreal numbers
Partially ordered sets


Dedekind cut
Generalizations
Arbitrary linearly ordered sets


Dedekind cut
Generalizations
Surreal numbers


Dedekind cut
Generalizations
Partially ordered sets


Regular representation, trace and discriminant


Algebraic number field
Definition
Prerequisites


Algebraic number field
Definition
Definition


Algebraic number field
Examples
Non-examples


Algebraic number field
Algebraicity, and ring of integers

Analytic objects: ζ-functions, L-functions, and class number formula


Algebraic number field
Algebraicity, and ring of integers
Unique factorization