Filter (set theory)
Filters and prefilters
Ultrafilters


Filter (set theory)
Filters and prefilters
Kernels

Classifying families by their kernels
Characterizing fixed ultra prefilters


Filter (set theory)
Filters and prefilters
Kernels
Classifying families by their kernels


Filter (set theory)
Filters and prefilters
Kernels
Characterizing fixed ultra prefilters


Finer/coarser, subordination, and meshing


Set theoretic properties and constructions

Trace and meshing
Images and preimages under functions
Products of prefilters
Set subtraction and some examples


Subordination is preserved by images and preimages


Nets to prefilters
Prefilters to nets
Subordinate filters and subnets


Filter (set theory)
Filters and nets
Nets to prefilters


Filter (set theory)
Filters and nets
Prefilters to nets


Filter (set theory)
Filters and nets
Prefilters to nets
Partially ordered net


Filter (set theory)
Filters and nets
Subordinate filters and subnets

Non–equivalence of subnets and subordinate filters


Non–equivalence of subnets and subordinate filters


Preliminaries
Pure identity theories
Unary relations
Equivalence relations
Orders
Lattices
Graphs
Boolean algebras
Groups
Rings and fields
Geometry
Differential algebra
Addition
Arithmetic
Second order arithmetic
Set theories


List of first-order theories
Preliminaries


List of first-order theories
Pure identity theories


List of first-order theories
Unary relations


List of first-order theories
Equivalence relations


List of first-order theories
Boolean algebras


List of first-order theories
Rings and fields


List of first-order theories
Differential algebra


List of first-order theories
Second order arithmetic


List of first-order theories
Set theories


Defining the ordered pair using set theory


Ordered pair
Informal and formal definitions


Defining the ordered pair using set theory

Wiener's definition
Hausdorff's definition
Kuratowski's definition
Quine–Rosser definition
Cantor–Frege definition
Morse definition
Axiomatic definition


Proving that definitions satisfy the characteristic property


Grothendieck universes and inaccessible cardinals


Intuition
Forcing conditions and forcing posets
Generic filters
P-names and interpretations
Forcing
Consistency
Cohen forcing
The countable chain condition
Easton forcing
Random reals
Boolean-valued models
Meta-mathematical explanation
Logical explanation


Forcing (mathematics)
Intuition
The role of the model


Forcing (mathematics)
Forcing conditions and forcing posets


Forcing (mathematics)
Forcing conditions and forcing posets
Examples


Forcing (mathematics)
P-names and interpretations


Forcing (mathematics)
P-names and interpretations
Rigorous definitions


Forcing (mathematics)
Forcing
Internal definition


Forcing (mathematics)
The countable chain condition


Forcing (mathematics)
Boolean-valued models


Forcing (mathematics)
Meta-mathematical explanation


Forcing (mathematics)
Logical explanation


Theory
Urelements in set theory
Quine atoms
References
External links


Predicate (mathematical logic)
Predicates in different systems


Ordinals extend the natural numbers
Definitions
Transfinite sequence
Transfinite induction
Arithmetic of ordinals
Ordinals and cardinals
Some "large" countable ordinals
Topology and ordinals
History


Ordinal number
Ordinals extend the natural numbers


Definition of an ordinal as an equivalence class


Ordinal number
Definitions
Well-ordered sets


Ordinal number
Definitions
Von Neumann definition of ordinals


Ordinal number
Definitions
Other definitions


Transfinite recursion
Successor and limit ordinals
Indexing classes of ordinals
Closed unbounded sets and classes


Ordinal number
Transfinite induction
Transfinite recursion


Ordinal number
Transfinite induction
Successor and limit ordinals


Ordinal number
Transfinite induction
Indexing classes of ordinals


Ordinal number
Transfinite induction
Closed unbounded sets and classes


Initial ordinal of a cardinal
Cofinality


Ordinal number
Ordinals and cardinals
Initial ordinal of a cardinal


Ordinal number
Ordinals and cardinals
Cofinality


Ordinal number
Some "large" countable ordinals


Well-definedness under an equivalence relation

Fundamental theorem of equivalence relations

Equivalence relations and mathematical logic


Alternative definition using relational algebra


Simple example
Equivalence relations
Relations that are not equivalences


Equivalence relation
Examples
Simple example


Equivalence relation
Examples
Equivalence relations


Equivalence relation
Examples
Relations that are not equivalences


Equivalence relation
Connections to other relations


Well-definedness under an equivalence relation


Equivalence relation
Related important definitions

Equivalence class
Quotient set
Projection
Equivalence kernel
Partition


Equivalence relation
Related important definitions
Equivalence class


Equivalence relation
Related important definitions
Quotient set


Equivalence relation
Related important definitions
Projection


Equivalence relation
Related important definitions
Equivalence kernel


Equivalence relation
Related important definitions
Partition


Equivalence relation
Related important definitions
Partition
Counting partitions


Fundamental theorem of equivalence relations


Equivalence relation
Comparing equivalence relations


Equivalence relation
Generating equivalence relations


Equivalence relation
Algebraic structure

Group theory
Categories and groupoids
Lattices


Equivalence relation
Algebraic structure
Group theory


Equivalence relation
Algebraic structure
Categories and groupoids


Equivalence relation
Algebraic structure
Lattices


Equivalence relations and mathematical logic


Necessary and sufficient conditions for finiteness


Finite set
Set-theoretic definitions of finiteness


Finite set
Set-theoretic definitions of finiteness
Other concepts of finiteness


Examples
Properties
Transitive closure
Transitive models of set theory


Transitive set
Transitive models of set theory


Inhabited set
Comparison with nonempty sets


Reasons for nonexistence
Theories of universality


Regularity
Russell's paradox
Cantor's theorem


Universal set
Reasons for nonexistence
Regularity


Universal set
Reasons for nonexistence
Russell's paradox


Universal set
Reasons for nonexistence
Cantor's theorem


Restricted comprehension
Universal objects that are not sets


Universal set
Theories of universality
Restricted comprehension


Universal set
Theories of universality
Universal objects that are not sets


Examples, properties, and sufficient conditions


Ultrafilter (set theory)
Generalization to ultra prefilters


Ultrafilter (set theory)
Characterizations

Grills and filter-grills
Free or principal


Ultrafilter (set theory)
Characterizations
Grills and filter-grills


Ultrafilter (set theory)
Characterizations
Free or principal


Ultrafilter (set theory)
The ultrafilter lemma

Relationships to other statements under ZF


Relationships to other statements under ZF

Statements that cannot be deduced
Equivalent statements
Weaker statements


Ultrafilter (set theory)
Ordering on ultrafilters


Ultrafilter (set theory)
Ultrafilters on ℘(ω)


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


Aircraft and automobiles
Cooling systems
Vacuum brakes


Inch of mercury
Applications
Aircraft and automobiles


Inch of mercury
Applications
Cooling systems


Inch of mercury
Applications
Vacuum brakes


Definition
Applications and interpretations
History
Philosophical perspectives


Finite and low cardinality stages of the hierarchy


Von Neumann universe
Applications and interpretations

Applications of V as models for set theories

Interpretation of V as the "set of all sets"


Applications of V as models for set theories


Interpretation of V as the "set of all sets"


Von Neumann universe
Applications and interpretations
V and the axiom of regularity


Von Neumann universe
Applications and interpretations
The existential status of V


Von Neumann universe
Philosophical perspectives


Definition
Inverse image
Notation for image and inverse image
Examples
Properties


Image of an element
Image of a subset
Image of a function
Generalization to binary relations


Image (mathematics)
Definition
Image of an element


Image (mathematics)
Definition
Image of a subset


Image (mathematics)
Definition
Image of a function


Image (mathematics)
Definition
Generalization to binary relations


Image (mathematics)
Notation for image and inverse image

Arrow notation
Star notation
Other terminology


Image (mathematics)
Notation for image and inverse image
Arrow notation


Image (mathematics)
Notation for image and inverse image
Star notation


Image (mathematics)
Notation for image and inverse image
Other terminology


General
Multiple functions
Multiple subsets of domain or codomain


Image (mathematics)
Properties
Multiple functions


Image (mathematics)
Properties
Multiple subsets of domain or codomain


Definition
Examples
Injections can be undone
Injections may be made invertible
Other properties
Proving that functions are injective
Gallery


Injective function
Injections can be undone


Injective function
Injections may be made invertible


Injective function
Proving that functions are injective


The generalized continuum hypothesis holds in L

Constructible sets are definable from the ordinals


Constructible universe
Additional facts about the sets Lα


Constructible universe
L is a standard inner model of ZFC


Constructible universe
L is absolute and minimal


Constructible universe
L is absolute and minimal
L and large cardinals


Constructible universe
L can be well-ordered


Constructible universe
L has a reflection principle


The generalized continuum hypothesis holds in L


Constructible sets are definable from the ordinals


Constructible universe
Relative constructibility


History
Geography
Demographics
Government and political geography
Economy
Transport
Media
Culture
Education
Health
Sports
Notable people from Oaxaca


Name
Prehistoric and pre-Hispanic period
Spanish colonization
Independence
Late 19th century to present


Oaxaca
History
Prehistoric and pre-Hispanic period


Oaxaca
History
Late 19th century to present


Regions, districts and major communities


Oaxaca
Geography
Nature and conservation

Lagunas de Chacahua National Park
Benito Juárez National Park
Huatulco National Park
Tehuacán-Cuicatlán Biosphere Reserve


Oaxaca
Geography
Nature and conservation
Lagunas de Chacahua National Park


Oaxaca
Geography
Nature and conservation
Benito Juárez National Park


Oaxaca
Geography
Nature and conservation
Huatulco National Park


Oaxaca
Geography
Nature and conservation
Tehuacán-Cuicatlán Biosphere Reserve


Largest cities
Overview
Indigenous peoples
Afro-Mexicans
Religion


Oaxaca
Demographics
Indigenous peoples
Zapotecs


Oaxaca
Demographics
Indigenous peoples
Mixtecs


Oaxaca
Demographics
Indigenous peoples
Mazatecos


Oaxaca
Demographics
Indigenous peoples
Mixe


Oaxaca
Demographics
Indigenous peoples
Other


Oaxaca
Government and political geography


Migration
Agriculture
Mining and industry
Commerce


Road, rail and sea
Air
Local transportation services


Oaxaca
Transport
Local transportation services


Arts
Food and drink
Landmarks and tourist attractions
Handicrafts
Symbols


Oaxaca
Culture
Landmarks and tourist attractions


Etymology
Geography
History
Economy
Culture
Education
Demographics
Tourism
Archeological sites
Government
Infrastructure
Major communities
Notable People


Political geography
Natural geography
Climate
Ecosystems


Pre-Columbian
Colonial period, 1519–1821
Independence
20th century to the present


Veracruz
History
Colonial period, 1519–1821


Veracruz
History
20th century to the present


Agriculture
Natural resources
Golden Lane Oil Fields
Industry, transportation and commerce
Handcrafts


Veracruz
Economy
Industry, transportation and commerce


Gastronomy
Museums
Fairs and festivals
Dance and music
Art and architecture
Literature
Religion


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


Geography
Demographics
Government and regions
History
Tourism
Culture
Economy
Education
Communication
Infrastructure
Sports
Twinning and covenants


Nomenclature and seal
Pre-Hispanic period
Colonial period
Independence
Mexican Revolution to present


Jalisco
History
Mexican Revolution to present


"Jalisco is Mexico"
Mariachi and other music
Traditional clothing and dance
Tequila
Cuisine
Catholic faith


Jalisco
Culture
Mariachi and other music


Jalisco
Culture
Traditional clothing and dance


Jalisco
Culture
Traditional clothing and dance
Dresses


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


British logistics in the Normandy campaign

Background
Planning
Assault
Build-up
Breakout and pursuit
Outcome
Notes
References


Organisation
Mulberry harbour
Minor ports
Ordnance
Subsistence
Construction
Salvage
Casualties
Provost


History Demographics Culture Tourism Handcrafts Politics Geography Economy Education Infrastructure


Independence and the later nineteenth century


Guerrero
History
Prehistoric and Pre-Columbian periods


Guerrero
History
Mexican Revolution to the present


Triangle of the Sun
Archeological sites
Outdoor activities


History
Phonology
Grammar
Writing system
Numbers of speakers
Status as a language
Influence on other languages
Language examples


Origins
Written evidence
Printing
Secularization
20th century


Consonants
Vowels
Dialectal variation
Comparison with German
Comparison with Hebrew


Yiddish
Phonology
Comparison with German


Yiddish
Phonology
Comparison with Hebrew


Israel and Zionism
South Africa
Former Soviet Union
Council of Europe
Sweden
United States
United Kingdom
Canada
Religious communities
Modern Yiddish education
Internet


Yiddish
Status as a language
Israel and Zionism


Yiddish
Status as a language
South Africa


Yiddish
Status as a language
Former Soviet Union


Yiddish
Status as a language
Former Soviet Union
Russia


Yiddish
Status as a language
Former Soviet Union
Ukraine


Yiddish
Status as a language
Council of Europe


Yiddish
Status as a language
United States


Yiddish
Status as a language
United States
Present U.S. speaker population


Yiddish
Status as a language
United Kingdom


Yiddish
Status as a language
Religious communities


Yiddish
Status as a language
Modern Yiddish education


Valentinus's revolt and Martinus's death


Martinus (son of Heraclius)
Life
Reign of Constantine III and Heraclonas


Martinus (son of Heraclius)
References
Citations


Martinus (son of Heraclius)
References
Bibliography


Martinus (son of Heraclius)
Primary sources


Early life
Playing career
International play
Personal life
Career statistics
Awards and honors
References
External links


Chris Kreider
Playing career
Professional


Chris Kreider
Playing career
Professional
Early years (2012–2015)


Chris Kreider
Playing career
Professional
Later years


Regular season and playoffs
International


Chris Kreider
Career statistics
Regular season and playoffs


Chris Kreider
Career statistics
International


Founding
Editors and contributors
Journalistic profile
Orthography
Decline and outbreak of war
References


Der Bialistoker Shtern
Editors and contributors


Der Bialistoker Shtern
Journalistic profile


Der Bialistoker Shtern
Decline and outbreak of war


Overview
Episodes
Notable incidents
Cancellation
Associated programs


Early life and family
Career
Death and legacy
Notes
References
External links


Sheikh-ul-Jamia, Jamia Millia Islamia (1926–1948)

Basic National Education Committee (1937)

Vice-Chancellor, Aligarh Muslim University (1948–1956)


Sheikh-ul-Jamia, Jamia Millia Islamia (1926–1948)


Basic National Education Committee (1937)


Vice-Chancellor, Aligarh Muslim University (1948–1956)


Zakir Husain
Career
Governor of Bihar (1957–1962)


Zakir Husain
Career
Vice President of India (1962–1967)


Zakir Husain
Career
President of India (1967–1969)


Zakir Husain
Death and legacy
Commemoration


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


History
Crest and colours
Grounds
Support
Global brand
Ownership and finances
Players
Player awards
Coaching staff
Management
Honours
Manchester United Women
Footnotes
References
Further reading
External links


Manchester United F.C.
History
Early years (1878–1945)


Manchester United F.C.
History
Busby years (1945–1969)


Manchester United F.C.
History
1969–1986


Manchester United F.C.
History
Ferguson years (1986–2013)


Manchester United F.C.
History
2013–present


Manchester United F.C.
Crest and colours


1878–1893: North Road
1893–1910: Bank Street
1910–present: Old Trafford


Manchester United F.C.
Grounds
1878–1893: North Road


Manchester United F.C.
Grounds
1893–1910: Bank Street


Manchester United F.C.
Grounds
1910–present: Old Trafford


Manchester United F.C.
Support
Rivalries


Manchester United F.C.
Global brand
Sponsorship


Manchester United F.C.
Ownership and finances


Manchester United F.C.
Ownership and finances
Glazer ownership


Manchester United F.C.
Players
First-team squad


Manchester United F.C.
Players
First-team squad
Out on loan


Manchester United F.C.
Players
Reserves and academy


Manchester United F.C.
Players
Reserves and academy
Out on loan


Manchester United F.C.
Coaching staff
Managerial history


Manchester United Limited
Manchester United Football Club


Manchester United F.C.
Management
Manchester United Limited


Manchester United F.C.
Management
Manchester United Football Club


Manchester United F.C.
Honours
Doubles and Trebles


Manchester United F.C.
Manchester United Women


Early life
Professional career
Pitching style
Post-playing career
Legacy
Personal life


Sandy Koufax
Early life
Major League tryouts


Early years (1955–1960)
Domination (1961–1964)
Playing in pain (1965–66)
Holdout
Retirement
Career overall


Sandy Koufax
Professional career
Early years (1955–1960)


Sandy Koufax
Professional career
Domination (1961–1964)

1961 season
1962 season
1963 season
1964 season


Sandy Koufax
Professional career
Domination (1961–1964)
1961 season


Sandy Koufax
Professional career
Domination (1961–1964)
1962 season


Sandy Koufax
Professional career
Domination (1961–1964)
1963 season


Sandy Koufax
Professional career
Domination (1961–1964)
1964 season


Sandy Koufax
Professional career
Playing in pain (1965–66)

1965 season
Perfection
World Series and Yom Kippur


Sandy Koufax
Professional career
Playing in pain (1965–66)
1965 season


Sandy Koufax
Professional career
Playing in pain (1965–66)
Perfection


Sandy Koufax
Professional career
Playing in pain (1965–66)
World Series and Yom Kippur


Sandy Koufax
Professional career
Holdout


Sandy Koufax
Professional career
Holdout
1966 season


Sandy Koufax
Professional career
Retirement


Sandy Koufax
Professional career
Career overall


Sandy Koufax
Professional career
Career overall
Career statistics


A&E Networks
History
Post-Lifetime acquisition


TV Channels
A&E Studios
A+E Networks International
A+E Networks Home Entertainment


Filmography
A&E IndieFilms
History Films


A&E Networks
Units
A&E Studios
Filmography


A&E Networks
Units
A&E Studios
A&E IndieFilms


A&E Networks
Units
A&E Studios
History Films


A&E Networks
Units
A+E Networks International

Availability
A+E Networks Latin America
A+E Networks UK
A+E Networks Germany
A+E Networks Italy
The History Channel Iberia
A+E Networks Asia
A+E Networks India


A&E Networks
Units
A+E Networks International
Availability


A&E Networks
Units
A+E Networks International
A+E Networks Latin America


A&E Networks
Units
A+E Networks International
A+E Networks UK


A&E Networks
Units
A+E Networks International
A+E Networks Germany


A&E Networks
Units
A+E Networks International
A+E Networks Italy


A&E Networks
Units
A+E Networks International
The History Channel Iberia


A&E Networks
Units
A+E Networks International
A+E Networks Asia


A&E Networks
Units
A+E Networks International
A+E Networks India


A&E Networks
Units
A+E Networks Home Entertainment


Selection process
Appointment
Powers and functions
Contingency situation
Emoluments
Removal
Legal immunity
Analysis of role in Government


Governor (India)
Selection process
Qualifications


Executive powers
Legislative powers
Financial powers
Discretionary powers


Governor (India)
Powers and functions
Executive powers


Governor (India)
Powers and functions
Legislative powers


Governor (India)
Powers and functions
Financial powers


Governor (India)
Powers and functions
Discretionary powers


Governor (India)
Analysis of role in Government


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


Early life
Business career
Writing career
Personal life
References
External links


Location
History
Functionality
Departments
Notes
References
External links


Republican Palace Museum
Location
Opening hours


Cultural and social activities
Educational activities


Republican Palace Museum
Functionality
Cultural and social activities


Republican Palace Museum
Functionality
Educational activities


Suite of musical instruments, utensils and furniture


Republican Palace Museum
Departments
Automobile suite


Republican Palace Museum
Departments
Photographs and paintings suite


Republican Palace Museum
Departments
Honours suite


Republican Palace Museum
Departments
Gift suite


Republican Palace Museum
Departments
Documentation suite


Republican Palace Museum
Departments
Decoration suite


Republican Palace Museum
Departments
Weapons suite


Republican Palace Museum
Departments
Library


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


Museums
Botanical gardens
Clubs
Places of worship


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


Background Destruction Attribution Consequences Reactions


Destruction of the Kakhovka Dam
Background


Destruction of the Kakhovka Dam
Destruction


Destruction of the Kakhovka Dam
Attribution

Claims of Russian responsibility
Claims of Ukrainian responsibility


Destruction of the Kakhovka Dam
Attribution
Claims of Russian responsibility


Destruction of the Kakhovka Dam
Attribution
Claims of Ukrainian responsibility


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
Ukraine


Destruction of the Kakhovka Dam
Reactions
International community


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


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


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


Kakhovka Dam
Russian invasion of Ukraine


Kakhovka Dam
Russian invasion of Ukraine
Destruction of the dam


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


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


Career
Personal life
Media appearances
Racing record
References
External links


Karting
Early formula car racing
IndyCar
Other racing


Josef Newgarden
Career
Early formula car racing

Skip Barber
Formula Ford
Formula Palmer Audi
GP3 Series
Indy Lights


Josef Newgarden
Career
Early formula car racing
Skip Barber


Josef Newgarden
Career
Early formula car racing
Formula Ford


Josef Newgarden
Career
Early formula car racing
Formula Palmer Audi


Josef Newgarden
Career
Early formula car racing
GP3 Series


Josef Newgarden
Career
Early formula car racing
Indy Lights


2012–2014: Sarah Fisher Hartman Racing
2015: CFH Racing
2016: Ed Carpenter Racing
2017–present: Team Penske


Josef Newgarden
Career
IndyCar
2012–2014: Sarah Fisher Hartman Racing


Josef Newgarden
Career
IndyCar
2015: CFH Racing


Josef Newgarden
Career
IndyCar
2016: Ed Carpenter Racing


Josef Newgarden
Career
IndyCar
2017–present: Team Penske


Complete IMSA SportsCar Championship results


Josef Newgarden
Racing record
Career summary


Josef Newgarden
Racing record
Complete GP3 Series results


Josef Newgarden
Racing record
American open–wheel racing results

Indy Lights
IndyCar Series
Indianapolis 500


Josef Newgarden
Racing record
American open–wheel racing results
Indy Lights


Josef Newgarden
Racing record
American open–wheel racing results
IndyCar Series


Josef Newgarden
Racing record
American open–wheel racing results
Indianapolis 500


Josef Newgarden
Racing record
Superstar Racing Experience


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


Crash
Emergency response
Victims
Investigation
Aftermath
Reactions


2023 Odisha train collision
Emergency response


2023 Odisha train collision
Investigation


Background
Road to the final
Match
References
Note
External links


2023 Indian Premier League final
Background


2023 Indian Premier League final
Road to the final

Group stage
League stage matches between finalists
Playoffs


2023 Indian Premier League final
Road to the final
Group stage


2023 Indian Premier League final
Road to the final
League stage matches between finalists


2023 Indian Premier League final
Road to the final
Playoffs


Match officials
Summary
Gujarat Titans innings
Chennai Super Kings innings
Scorecard


2023 Indian Premier League final
Match
Match officials


2023 Indian Premier League final
Match
Summary


2023 Indian Premier League final
Match
Gujarat Titans innings


2023 Indian Premier League final
Match
Chennai Super Kings innings


2023 Indian Premier League final
Match
Scorecard


2023 Indian Premier League final
References


2023 Indian Premier League final
External links


Songs written by Barry Mann and Cynthia Weil


Events
Births
Deaths
Holidays and observances
References
External links


Early life
Education
Career
Writings
Death
Honours and recognition
Selected works
References
Further reading
External links


Biography
Style and subject matter
References
External links


Natasha Al-Maani
Style and subject matter


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


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


Background
Timeline
Casualties
Foreign involvement
Evacuation of foreign nationals
Humanitarian impact
Peace efforts
Disinformation
Reactions
References


2023 Sudan conflict
Background
Political context


By location
Foreign casualties
Casualties among humanitarian workers


2023 Sudan conflict
Casualties
By location


2023 Sudan conflict
Casualties
Foreign casualties


2023 Sudan conflict
Casualties
Casualties among humanitarian workers


Libya
Wagner Group
Egypt
Ethiopia
United Arab Emirates


2023 Sudan conflict
Foreign involvement
Libya


2023 Sudan conflict
Foreign involvement
Wagner Group


2023 Sudan conflict
Foreign involvement
Egypt


2023 Sudan conflict
Foreign involvement
Egypt
Egyptian POWs


2023 Sudan conflict
Foreign involvement
Ethiopia


2023 Sudan conflict
Foreign involvement
United Arab Emirates


2023 Sudan conflict
Evacuation of foreign nationals


2023 Sudan conflict
Humanitarian impact
Refugees


2023 Sudan conflict
Reactions
Domestic
Military


2023 Sudan conflict
Reactions
Domestic
Civilian


2023 Sudan conflict
Reactions
International


2023 Sudan conflict
Reactions
International
Countries


2023 Sudan conflict
Reactions
International
Organizations


Career
Recordings
Publication
Awards
References
External links


Early life and education
Career
Death
Selected filmography
Books
Notes
References
External links


Jessie Maple
Selected filmography
Features


Jessie Maple
Selected filmography
Documentaries


Early life
College career
Professional career
References


June
Previous months
References
External links


Induced substructures and closed subsets


Structure (mathematical logic)
Definition

Domain
Signature
Interpretation function
Examples


Structure (mathematical logic)
Definition
Domain


Structure (mathematical logic)
Definition
Signature


Structure (mathematical logic)
Definition
Interpretation function


Structure (mathematical logic)
Definition
Examples


Structure (mathematical logic)
Homomorphisms and embeddings

Homomorphisms
Embeddings
Example
Homomorphism problem


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphisms


Structure (mathematical logic)
Homomorphisms and embeddings
Embeddings


Structure (mathematical logic)
Homomorphisms and embeddings
Example


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphism problem


Structure (mathematical logic)
Structures and first-order logic

Satisfaction relation
Definable relations


Structure (mathematical logic)
Structures and first-order logic
Satisfaction relation


Structure (mathematical logic)
Structures and first-order logic
Definable relations

Definability with parameters
Implicit definability


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Definability with parameters


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Implicit definability


Structure (mathematical logic)
Many-sorted structures


Structure (mathematical logic)
Other generalizations

Partial algebras
Structures for typed languages
Higher-order languages
Structures that are proper classes


Structure (mathematical logic)
Other generalizations
Partial algebras


Structure (mathematical logic)
Other generalizations
Structures for typed languages


Structure (mathematical logic)
Other generalizations
Higher-order languages


Structure (mathematical logic)
Other generalizations
Structures that are proper classes


History
Military base
Nature
Education
Population
Demographics
International relations
Notable residents


Keila
International relations
Twin towns — sister cities


Life cycle of Cladius difformis
Taxonomy
References
Bibliography
External links


Tenthredinidae
Life cycle of Cladius difformis


Tenthredinidae
Taxonomy
Subfamilies and genera


Events
Births
Deaths
Holidays and observances
References
External links


Description
Cultivation
Other uses
Select species


Distribution and habitat
Description
Biology
Gallery
References
External links


Tenthredo scrophulariae
Distribution and habitat


Definition
Conceptions of deduction
Rules of inference
Validity and soundness
Difference from ampliative reasoning
In various fields
History
Related concepts and theories


Deductive reasoning
Conceptions of deduction


Prominent rules of inference
Fallacies
Definitory and strategic rules


Deductive reasoning
Rules of inference
Prominent rules of inference

Modus ponens
Modus tollens
Hypothetical syllogism


Deductive reasoning
Rules of inference
Prominent rules of inference
Modus ponens


Deductive reasoning
Rules of inference
Prominent rules of inference
Modus tollens


Deductive reasoning
Rules of inference
Prominent rules of inference
Hypothetical syllogism


Deductive reasoning
Rules of inference
Fallacies


Deductive reasoning
Rules of inference
Definitory and strategic rules


Deductive reasoning
Validity and soundness


Deductive reasoning
Difference from ampliative reasoning


Cognitive psychology
Epistemology
Probability logic


Deductive reasoning
In various fields
Cognitive psychology

Psychological theories of deductive reasoning


Psychological theories of deductive reasoning


Deductive reasoning
In various fields
Cognitive psychology
Intelligence


Deductive reasoning
In various fields
Epistemology


Deductive reasoning
In various fields
Probability logic


Deductive reasoning
Related concepts and theories

Deductivism
Natural deduction
Geometrical method


Deductive reasoning
Related concepts and theories
Deductivism


Deductive reasoning
Related concepts and theories
Natural deduction


Deductive reasoning
Related concepts and theories
Geometrical method


Mission
History
Projects and initiatives
Technology
Finances
Staff
Independent contractors
Disputes
References
External links


Wikimedia Foundation
Projects and initiatives


Wikimedia Foundation
Projects and initiatives
Wikimedia projects


Wikimedia Foundation
Projects and initiatives
Affiliates


Wikimedia Foundation
Projects and initiatives
Wikimania


Wikimedia Foundation
Technology
Hardware


Wikimedia Foundation
Technology
Software


Wikimedia Endowment
Financial development
Expenses
Grants


Wikimedia Foundation
Finances
Wikimedia Endowment


Wikimedia Foundation
Finances
Financial development


History
Present department structure
Board of trustees


Wikimedia Foundation
Staff
Present department structure


Wikimedia Foundation
Staff
Board of trustees


Wikimedia Foundation
Independent contractors


Excessive spending and fundraising
Knowledge Engine project


Wikimedia Foundation
Disputes
Excessive spending and fundraising


Wikimedia Foundation
Disputes
Knowledge Engine project


Organization
Financials
Charity status
Community


Wikimedia Foundation
External links
Organization


Wikimedia Foundation
External links
Financials


Wikimedia Foundation
External links
Charity status


Wikimedia Foundation
External links
Community


Notable theories on the structure of concepts


Concepts as mental representations
Concepts as abstract objects


Concept
Ontology of concepts
Concepts as mental representations

The psychological view of concepts
The physicalist view of concepts


Concept
Ontology of concepts
Concepts as mental representations
The psychological view of concepts


Concept
Ontology of concepts
Concepts as mental representations
The physicalist view of concepts


Concept
Ontology of concepts
Concepts as abstract objects

A priori concepts
Embodied content
Realist universal concepts
Sense and reference
Concepts in calculus


Concept
Ontology of concepts
Concepts as abstract objects
A priori concepts


Concept
Ontology of concepts
Concepts as abstract objects
Embodied content


Concept
Ontology of concepts
Concepts as abstract objects
Realist universal concepts


Concept
Ontology of concepts
Concepts as abstract objects
Sense and reference


Concept
Ontology of concepts
Concepts as abstract objects
Concepts in calculus


Notable theories on the structure of concepts

Classical theory
Prototype theory
Theory-theory


Etymology
Overview
Types of ontologies
History
Other ontological topics


Particulars and universals
Abstract and concrete
Ontological dependence
Identity
Modality
Substances
Properties and relations
States of affairs and events
Reality of things


Ontology
Overview
Particulars and universals


Ontology
Overview
Ontological dependence


Ontology
Overview
Properties and relations


Ontology
Overview
States of affairs and events


Information science and natural sciences


Ontology
Types of ontologies
Flat vs polycategorical vs hierarchical


Ontology
Types of ontologies
Thing ontologies vs fact ontologies


Ontology
Types of ontologies
Constituent ontologies vs blob theories


Ancient Greek
Hindu philosophy
Medieval
Modern
20th century


Ontological and epistemological certainty

Body and environment, questioning the meaning of being


Ontology
Other ontological topics
Ontological formations


Ontology
Other ontological topics
Ontology of fictional characters


Ontological and epistemological certainty


Body and environment, questioning the meaning of being


Ontology
Other ontological topics
Ontology and language


Ontology
Other ontological topics
Ontology and human geography


Ontology
Other ontological topics
Anthropology


Ontology
Other ontological topics
Reality and actuality


Ontology
Other ontological topics
Microcosmic ontology


Ontology
Other ontological topics
Ontological argument


Ontology
Other ontological topics
Hintikka's locution for existence


History
Main classification
Subclasses of the integers
Subclasses of the complex numbers
Extensions of the concept


Numerals
First use of numbers
Zero
Negative numbers
Rational numbers
Irrational numbers
Transcendental numbers and reals
Infinity and infinitesimals
Complex numbers
Prime numbers


Number
History
Transcendental numbers and reals


Number
History
Infinity and infinitesimals


Natural numbers
Integers
Rational numbers
Real numbers
Complex numbers


Number
Main classification
Natural numbers


Number
Main classification
Rational numbers


Number
Main classification
Complex numbers


Even and odd numbers
Prime numbers
Other classes of integers


Number
Subclasses of the integers
Even and odd numbers


Number
Subclasses of the integers
Prime numbers


Number
Subclasses of the integers
Other classes of integers


Number
Subclasses of the complex numbers

Algebraic, irrational and transcendental numbers


Algebraic, irrational and transcendental numbers


Number
Subclasses of the complex numbers
Constructible numbers


Number
Subclasses of the complex numbers
Computable numbers


p-adic numbers
Hypercomplex numbers
Transfinite numbers
Nonstandard numbers


Number
Extensions of the concept
p-adic numbers


Number
Extensions of the concept
Hypercomplex numbers


Number
Extensions of the concept
Transfinite numbers


Number
Extensions of the concept
Nonstandard numbers


Partial transformations
Algebraic structures
Combinatorics


Transformation (function)
Partial transformations


Transformation (function)
Algebraic structures


History
Taxonomy of spaces
Types of spaces
Spaces and structure
List of mathematical spaces


The golden age of geometry and afterwards


Space (mathematics)
History
Before the golden age of geometry


Three taxonomic ranks
Relations between species of spaces


Space (mathematics)
Taxonomy of spaces
Three taxonomic ranks


Space (mathematics)
Taxonomy of spaces
Relations between species of spaces


Normed, Banach, inner product, and Hilbert spaces

Measurable, measure, and probability spaces


Space (mathematics)
Types of spaces
Linear and topological spaces


Space (mathematics)
Types of spaces
Affine and projective spaces


Space (mathematics)
Types of spaces
Metric and uniform spaces


Normed, Banach, inner product, and Hilbert spaces


Space (mathematics)
Types of spaces
Smooth and Riemannian manifolds


Measurable, measure, and probability spaces


Space (mathematics)
Types of spaces
Non-commutative geometry


Space (mathematics)
Types of spaces
Schemes


Space (mathematics)
Types of spaces
Topoi


Space (mathematics)
Spaces and structure


Space (mathematics)
List of mathematical spaces


History
Main subdivisions
Other subfields
Applications
Prizes


Origins
Early modern number theory
Maturity and division into subfields


Classical Greece and the early Hellenistic period


Number theory
History
Origins
Dawn of arithmetic


Number theory
History
Origins
Diophantus


Number theory
History
Origins
Āryabhaṭa, Brahmagupta, Bhāskara


Number theory
History
Origins
Arithmetic in the Islamic golden age


Number theory
History
Origins
Western Europe in the Middle Ages


Number theory
History
Early modern number theory

Fermat
Euler
Lagrange, Legendre, and Gauss


Number theory
History
Early modern number theory
Fermat


Number theory
History
Early modern number theory
Euler


Number theory
History
Early modern number theory
Lagrange, Legendre, and Gauss


Number theory
History
Maturity and division into subfields


Elementary number theory
Analytic number theory
Algebraic number theory
Diophantine geometry


Number theory
Main subdivisions
Elementary number theory


Number theory
Main subdivisions
Analytic number theory


Number theory
Main subdivisions
Algebraic number theory


Number theory
Main subdivisions
Diophantine geometry


Probabilistic number theory
Arithmetic combinatorics
Computational number theory


Number theory
Other subfields
Probabilistic number theory


Number theory
Other subfields
Arithmetic combinatorics


Number theory
Other subfields
Computational number theory


Approaches and subfields of combinatorics


Approaches and subfields of combinatorics

Enumerative combinatorics
Analytic combinatorics
Partition theory
Graph theory
Design theory
Finite geometry
Order theory
Matroid theory
Extremal combinatorics
Probabilistic combinatorics
Algebraic combinatorics
Combinatorics on words
Geometric combinatorics
Topological combinatorics
Arithmetic combinatorics
Infinitary combinatorics


Combinatorial optimization
Coding theory
Discrete and computational geometry
Combinatorics and dynamical systems
Combinatorics and physics


Combinatorics
Related fields
Combinatorial optimization


Combinatorics
Related fields
Coding theory


Combinatorics
Related fields
Discrete and computational geometry


Combinatorics
Related fields
Combinatorics and dynamical systems


Combinatorics
Related fields
Combinatorics and physics


Point masses and the Dirac delta function


Point (geometry)
Points in Euclidean geometry


Vector space dimension
Topological dimension
Hausdorff dimension


Point (geometry)
Dimension of a point
Vector space dimension


Point (geometry)
Dimension of a point
Topological dimension


Point (geometry)
Dimension of a point
Hausdorff dimension


Point (geometry)
Geometry without points


Etymology
Classification
Properties and formulas
Generalizations
Naming
History
In nature
Computer graphics


Number of sides
Convexity and intersection
Equality and symmetry
Miscellaneous


Polygon
Classification
Convexity and intersection


Polygon
Classification
Equality and symmetry


Simple polygons
Regular polygons
Self-intersecting


Polygon
Properties and formulas
Area
Simple polygons


Polygon
Properties and formulas
Area
Regular polygons


Polygon
Properties and formulas
Area
Self-intersecting


Polygon
Properties and formulas
Centroid


Properties
Definition
Other representations
Generalizations of the Euclidean line
Extensions


In higher dimensions
Types
In axiomatic systems


Line (geometry)
Properties
In higher dimensions


Line (geometry)
Properties
In higher dimensions
Collinear points


Line (geometry)
Properties
In axiomatic systems


Linear equation
Parametric equation
Hesse normal form


Line (geometry)
Definition
Linear equation


Line (geometry)
Definition
Parametric equation


Line (geometry)
Definition
Hesse normal form


Line (geometry)
Other representations
Vectors


Line (geometry)
Other representations
Polar coordinates


Line (geometry)
Generalizations of the Euclidean line


Line (geometry)
Generalizations of the Euclidean line
Projective geometry


Growth of number of derangements as n approaches ∞


Derivation by inclusion–exclusion principle


Growth of number of derangements as n approaches ∞

Asymptotic expansion in terms of Bell numbers


Asymptotic expansion in terms of Bell numbers


Points, lines, and circles associated with a triangle

Further formulas for general Euclidean triangles

Specifying the location of a point in a triangle


Triangle
Types of triangle
By lengths of sides


Triangle
Types of triangle
By internal angles


Similarity and congruence
Right triangles


Triangle
Basic facts
Similarity and congruence


Condition on the sides
Conditions on the angles


Triangle
Existence of a triangle
Condition on the sides


Triangle
Existence of a triangle
Conditions on the angles


Triangle
Existence of a triangle
Conditions on the angles
Trigonometric conditions


Points, lines, and circles associated with a triangle


Trigonometric ratios in right triangles
Sine, cosine and tangent rules
Solution of triangles


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles

Sine, cosine and tangent
Inverse functions


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles
Sine, cosine and tangent


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles
Inverse functions


Triangle
Computing the sides and angles
Sine, cosine and tangent rules


Triangle
Computing the sides and angles
Solution of triangles


Further formulas for general Euclidean triangles

Medians, angle bisectors, perpendicular side bisectors, and altitudes


Medians, angle bisectors, perpendicular side bisectors, and altitudes


Triangle
Figures inscribed in a triangle

Conics
Convex polygon
Hexagon
Squares
Triangles


Triangle
Figures inscribed in a triangle
Conics


Triangle
Figures inscribed in a triangle
Convex polygon


Triangle
Figures inscribed in a triangle
Hexagon


Triangle
Figures inscribed in a triangle
Squares


Triangle
Figures inscribed in a triangle
Triangles


Triangle
Figures circumscribed about a triangle


Specifying the location of a point in a triangle


Probability: sampling a random combination


Example of counting combinations
Enumerating k-combinations


Combination
Number of k-combinations
Example of counting combinations


Combination
Number of k-combinations
Enumerating k-combinations


Combination
Number of combinations with repetition


Combination
Number of combinations with repetition
Example of counting multisubsets


Combination
Number of k-combinations for all k


Combination
Number of ways to put objects into bins


Characterizations
Properties
Coordinates and equations
Construction
Symmetry
Squares inscribed in triangles
Squaring the circle
Non-Euclidean geometry
Crossed square
Graphs


Regular pentagons
Equilateral pentagons
Cyclic pentagons
General convex pentagons
Pentagons in tiling
Pentagons in polyhedra
Pentagons in nature
Other examples


Chords from the circumscribed circle to the vertices


Pentagon
Regular pentagons
Derivation of the area formula


Pentagon
Regular pentagons
Point in plane


Pentagon
Regular pentagons
Geometrical constructions

Richmond's method
Carlyle circles
Euclid's method


Pentagon
Regular pentagons
Geometrical constructions
Richmond's method


Pentagon
Regular pentagons
Geometrical constructions
Carlyle circles


Pentagon
Regular pentagons
Geometrical constructions
Euclid's method


Pentagon
Regular pentagons
Physical construction methods


Pentagon
Regular pentagons
Regular pentagram


Heuristic mathematics and experimental mathematics


Mathematical proof
History and etymology


Direct proof
Proof by mathematical induction
Proof by contraposition
Proof by contradiction
Proof by construction
Proof by exhaustion
Probabilistic proof
Combinatorial proof
Nonconstructive proof
Statistical proofs in pure mathematics
Computer-assisted proofs


Mathematical proof
Methods of proof
Direct proof


Mathematical proof
Methods of proof
Proof by mathematical induction


Mathematical proof
Methods of proof
Proof by contraposition


Mathematical proof
Methods of proof
Proof by contradiction


Mathematical proof
Methods of proof
Proof by construction


Mathematical proof
Methods of proof
Proof by exhaustion


Mathematical proof
Methods of proof
Probabilistic proof


Mathematical proof
Methods of proof
Combinatorial proof


Mathematical proof
Methods of proof
Nonconstructive proof


Mathematical proof
Methods of proof
Statistical proofs in pure mathematics


Mathematical proof
Methods of proof
Computer-assisted proofs


Mathematical proof
Undecidable statements


Inductive logic proofs and Bayesian analysis

Influence of mathematical proof methods outside mathematics


Mathematical proof
Related concepts
Visual proof


Mathematical proof
Related concepts
Elementary proof


Mathematical proof
Related concepts
Two-column proof


Mathematical proof
Related concepts
Colloquial use of "mathematical proof"


Mathematical proof
Related concepts
Statistical proof using data


Inductive logic proofs and Bayesian analysis


Mathematical proof
Related concepts
Proofs as mental objects


Influence of mathematical proof methods outside mathematics


Mathematical proof
History and etymology


Mathematical proof
Methods of proof
Direct proof


Mathematical proof
Methods of proof
Proof by mathematical induction


Mathematical proof
Methods of proof
Proof by contraposition


Mathematical proof
Methods of proof
Proof by contradiction


Mathematical proof
Methods of proof
Proof by construction


Mathematical proof
Methods of proof
Proof by exhaustion


Mathematical proof
Methods of proof
Probabilistic proof


Mathematical proof
Methods of proof
Combinatorial proof


Mathematical proof
Methods of proof
Nonconstructive proof


Mathematical proof
Methods of proof
Statistical proofs in pure mathematics


Mathematical proof
Methods of proof
Computer-assisted proofs


Mathematical proof
Undecidable statements


Heuristic mathematics and experimental mathematics


Mathematical proof
Related concepts
Visual proof


Mathematical proof
Related concepts
Elementary proof


Mathematical proof
Related concepts
Two-column proof


Mathematical proof
Related concepts
Colloquial use of "mathematical proof"


Mathematical proof
Related concepts
Statistical proof using data


Inductive logic proofs and Bayesian analysis


Mathematical proof
Related concepts
Proofs as mental objects


Influence of mathematical proof methods outside mathematics


History
Main concepts
Contemporary geometry
Applications


Axioms
Objects
Length, area, and volume
Congruence and similarity
Compass and straightedge constructions
Dimension
Symmetry


Points Lines Planes Angles Curves Surfaces Manifolds


Geometry
Main concepts
Objects
Manifolds


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


Equilateral triangles on the sides of an arbitrary hexagon

Gallery of natural and artificial hexagons


Hexagon
Related polygons and tilings
Self-crossing hexagons


Hexagon
Hexagon inscribed in a conic section


Hexagon
Hexagon inscribed in a conic section
Cyclic hexagon


Hexagon
Hexagon tangential to a conic section


Equilateral triangles on the sides of an arbitrary hexagon


Hexagon
Convex equilateral hexagon
Polyhedra with hexagons


Gallery of natural and artificial hexagons


A geometrical construction to find a sector area

Focal length and radius of curvature at the vertex

As the affine image of the unit parabola


Parabola
Definition as a locus of points


Parabola
In a Cartesian coordinate system

Axis of symmetry parallel to the y axis
General position


Parabola
In a Cartesian coordinate system
Axis of symmetry parallel to the y axis


Parabola
In a Cartesian coordinate system
General position


Parabola
Similarity to the unit parabola


Parabola
Conic section and quadratic form

Diagram, description, and definitions
Derivation of quadratic equation
Focal length
Position of the focus
Alternative proof with Dandelin spheres


Parabola
Conic section and quadratic form
Diagram, description, and definitions


Parabola
Conic section and quadratic form
Derivation of quadratic equation


Parabola
Conic section and quadratic form
Focal length


Parabola
Conic section and quadratic form
Position of the focus


Parabola
Conic section and quadratic form
Alternative proof with Dandelin spheres


Parabola
Proof of the reflective property

Construction and definitions
Deductions
Other consequences
Alternative proofs


Parabola
Proof of the reflective property
Construction and definitions


Parabola
Proof of the reflective property
Deductions


Parabola
Proof of the reflective property
Other consequences

Intersection of a tangent and perpendicular from focus

Reflection of light striking the convex side


Parabola
Proof of the reflective property
Other consequences
Tangent bisection property


Intersection of a tangent and perpendicular from focus


Reflection of light striking the convex side


Parabola
Proof of the reflective property
Alternative proofs


Parabola
Properties related to Pascal's theorem

4-points property
3-points–1-tangent property
2-points–2-tangents property
Axis direction


Parabola
Properties related to Pascal's theorem
4-points property


Parabola
Properties related to Pascal's theorem
3-points–1-tangent property


Parabola
Properties related to Pascal's theorem
2-points–2-tangents property


Parabola
Properties related to Pascal's theorem
Axis direction


Parabola
Steiner generation
Dual parabola


Parabola
Inscribed angles and the 3-point form


Two tangent properties related to the latus rectum


Parabola
Tangent properties
Orthoptic property


Parabola
Tangent properties
Lambert's theorem


Parabola
Facts related to chords and arcs

Focal length calculated from parameters of a chord

Area enclosed between a parabola and a chord

Corollary concerning midpoints and endpoints of chords


Focal length calculated from parameters of a chord


Area enclosed between a parabola and a chord


Corollary concerning midpoints and endpoints of chords


Parabola
Facts related to chords and arcs
Arc length


A geometrical construction to find a sector area


Focal length and radius of curvature at the vertex


As the affine image of the unit parabola


Regular octahedron
Irregular octahedra
Octahedra in the physical world
Related polyhedra


Dimensions
Orthogonal projections
Spherical tiling
Cartesian coordinates
Area and volume
Geometric relations
Characteristic orthoscheme
Topology
Nets
Faceting
Uniform colorings and symmetry


Octahedron
Regular octahedron
Dimensions


Octahedron
Regular octahedron
Orthogonal projections


Octahedron
Regular octahedron
Spherical tiling


Octahedron
Regular octahedron
Cartesian coordinates


Octahedron
Regular octahedron
Area and volume


Octahedron
Regular octahedron
Geometric relations

Dual
Stellation
Snub octahedron
Tessellations


Octahedron
Regular octahedron
Geometric relations
Dual


Octahedron
Regular octahedron
Geometric relations
Stellation


Octahedron
Regular octahedron
Geometric relations
Snub octahedron


Octahedron
Regular octahedron
Geometric relations
Tessellations


Octahedron
Regular octahedron
Characteristic orthoscheme


Octahedron
Regular octahedron
Uniform colorings and symmetry


Octahedron
Octahedra in the physical world

Octahedra in nature
Octahedra in art and culture
Tetrahedral octet truss


Octahedron
Octahedra in the physical world
Octahedra in nature


Octahedron
Octahedra in the physical world
Octahedra in art and culture


Octahedron
Octahedra in the physical world
Tetrahedral octet truss


Tetratetrahedron
Trigonal antiprism
Square bipyramid
Other related polyhedra


Octahedron
Related polyhedra
Tetratetrahedron


Octahedron
Related polyhedra
Trigonal antiprism


Octahedron
Related polyhedra
Square bipyramid


Octahedron
Related polyhedra
Other related polyhedra


Definition as locus of points
In Cartesian coordinates
Parametric representation
Polar forms
Eccentricity and the directrix property
Focus-to-focus reflection property
Conjugate diameters
Orthogonal tangents
Drawing ellipses
Inscribed angles and three-point form
Pole-polar relation
Metric properties
In triangle geometry
As plane sections of quadrics
Applications


Standard equation
Parameters
Tangent
Shifted ellipse
General ellipse


Ellipse
In Cartesian coordinates
Standard equation


Ellipse
In Cartesian coordinates
Parameters

Principal axes
Linear eccentricity
Eccentricity
Semi-latus rectum


Ellipse
In Cartesian coordinates
Parameters
Principal axes


Ellipse
In Cartesian coordinates
Parameters
Linear eccentricity


Ellipse
In Cartesian coordinates
Parameters
Eccentricity


Ellipse
In Cartesian coordinates
Parameters
Semi-latus rectum


Ellipse
In Cartesian coordinates
Tangent


Ellipse
In Cartesian coordinates
Shifted ellipse


Ellipse
In Cartesian coordinates
General ellipse


Standard parametric representation
Rational representation
Tangent slope as parameter
General ellipse


Ellipse
Parametric representation
Standard parametric representation


Ellipse
Parametric representation
Rational representation


Ellipse
Parametric representation
Tangent slope as parameter


Ellipse
Parametric representation
General ellipse


Polar form relative to center
Polar form relative to focus


Ellipse
Polar forms
Polar form relative to center


Ellipse
Polar forms
Polar form relative to focus


Ellipse
Eccentricity and the directrix property


Ellipse
Focus-to-focus reflection property


Theorem of Apollonios on conjugate diameters


Ellipse
Conjugate diameters
Definition of conjugate diameters


de La Hire's point construction
Pins-and-string method
Paper strip methods
Approximation by osculating circles
Steiner generation
As hypotrochoid


Ellipse
Drawing ellipses
de La Hire's point construction


Ellipse
Drawing ellipses
Pins-and-string method


Ellipse
Drawing ellipses
Paper strip methods


Ellipse
Drawing ellipses
Approximation by osculating circles


Ellipse
Drawing ellipses
Steiner generation


Ellipse
Drawing ellipses
As hypotrochoid


Ellipse
Inscribed angles and three-point form


Ellipse
Inscribed angles and three-point form
Circles

Inscribed angle theorem for circles
Three-point form of circle equation


Ellipse
Inscribed angles and three-point form
Circles
Inscribed angle theorem for circles


Ellipse
Inscribed angles and three-point form
Circles
Three-point form of circle equation


Ellipse
Inscribed angles and three-point form
Ellipses

Inscribed angle theorem for ellipses
Three-point form of ellipse equation


Ellipse
Inscribed angles and three-point form
Ellipses
Inscribed angle theorem for ellipses


Ellipse
Inscribed angles and three-point form
Ellipses
Three-point form of ellipse equation


Physics
Statistics and finance
Computer graphics
Optimization theory


Elliptical reflectors and acoustics
Planetary orbits
Harmonic oscillators
Phase visualization
Elliptical gears
Optics


Ellipse
Applications
Physics
Elliptical reflectors and acoustics


Ellipse
Applications
Physics
Planetary orbits


Ellipse
Applications
Physics
Harmonic oscillators


Ellipse
Applications
Physics
Phase visualization


Ellipse
Applications
Physics
Elliptical gears


Ellipse
Applications
Statistics and finance


Ellipse
Applications
Optimization theory


Regular dodecahedron
Other pentagonal dodecahedra
Rhombic dodecahedron
Other dodecahedra
Practical usage


Dodecahedron
Other pentagonal dodecahedra

Pyritohedron
Tetartoid
Dual of triangular gyrobianticupola


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron

Crystal pyrite
Cartesian coordinates
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Crystal pyrite


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Cartesian coordinates


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Tetartoid


Dodecahedron
Other pentagonal dodecahedra
Tetartoid
Cartesian coordinates


Dodecahedron
Other pentagonal dodecahedra
Tetartoid
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Dual of triangular gyrobianticupola


Definition
Characteristics
Convex polyhedra
Symmetries
Other important families of polyhedra
Generalisations of polyhedra
Alternative usages
History
In nature


Number of faces
Topological classification
Duality
Vertex figures
Surface area and distances
Volume
Dehn invariant


Polyhedron
Characteristics
Number of faces


Polyhedron
Characteristics
Topological classification


Polyhedron
Characteristics
Vertex figures


Polyhedron
Characteristics
Surface area and distances


Polyhedron
Characteristics
Dehn invariant


Regular polyhedra
Uniform polyhedra and their duals
Isohedra
Symmetry groups


Polyhedron
Symmetries
Uniform polyhedra and their duals


Polyhedron
Other important families of polyhedra

Polyhedra with regular faces
Pyramids
Stellations and facettings
Zonohedra
Space-filling polyhedra
Lattice polyhedra
Flexible polyhedra
Compounds
Orthogonal polyhedra
Embedded regular maps with planar faces
Canonical polyhedra


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces
Equal regular faces


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces
Johnson solids


Polyhedron
Other important families of polyhedra
Pyramids


Polyhedron
Other important families of polyhedra
Stellations and facettings


Polyhedron
Other important families of polyhedra
Zonohedra


Polyhedron
Other important families of polyhedra
Space-filling polyhedra


Polyhedron
Other important families of polyhedra
Lattice polyhedra


Polyhedron
Other important families of polyhedra
Flexible polyhedra


Polyhedron
Other important families of polyhedra
Compounds


Polyhedron
Other important families of polyhedra
Orthogonal polyhedra


Polyhedron
Other important families of polyhedra
Embedded regular maps with planar faces


Polyhedron
Other important families of polyhedra
Canonical polyhedra


Apeirohedra
Complex polyhedra
Curved polyhedra
Ideal polyhedra
Skeletons and polyhedra as graphs


Polyhedron
Generalisations of polyhedra
Apeirohedra


Polyhedron
Generalisations of polyhedra
Complex polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra

Spherical polyhedra
Curved spacefilling polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra
Spherical polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra
Curved spacefilling polyhedra


Polyhedron
Generalisations of polyhedra
Ideal polyhedra


Polyhedron
Generalisations of polyhedra
Skeletons and polyhedra as graphs


Higher-dimensional polyhedra
Topological polyhedra
Abstract polyhedra


Polyhedron
Alternative usages
Higher-dimensional polyhedra


Polyhedron
Alternative usages
Topological polyhedra


Polyhedron
Alternative usages
Abstract polyhedra


Before the Greeks
Ancient Greece
Ancient China
Medieval Islam
Renaissance
17th–19th centuries
20th–21st centuries


Basic terminology
Equations
Properties
Treatment by area of mathematics
Curves on a sphere
Generalizations
History
Gallery
Regions


Enclosed volume
Surface area
Other geometric properties
Properties of the sphere


Sphere
Properties
Other geometric properties


Sphere
Properties
Other geometric properties
Pencil of spheres


Sphere
Properties
Properties of the sphere


Spherical geometry
Differential geometry
Topology


Sphere
Treatment by area of mathematics
Spherical geometry


Sphere
Treatment by area of mathematics
Differential geometry


Sphere
Treatment by area of mathematics
Topology


Intersection of a sphere with a more general surface


Sphere
Curves on a sphere
Spherical conics


Inscription in or circumscription about other figures


Circumference
Area enclosed
Equations
Tangent lines


Cartesian coordinates
Polar coordinates
Complex plane


Circle
Analytic results
Equations
Cartesian coordinates


Circle
Analytic results
Equations
Polar coordinates


Circle
Analytic results
Equations
Complex plane


Chord
Tangent
Theorems
Inscribed angles
Sagitta


Circle
Compass and straightedge constructions

Construction through three noncollinear points


Circle
Compass and straightedge constructions
Construction with given diameter


Construction through three noncollinear points


Circle
Circle of Apollonius
Cross-ratios


Circle
Circle of Apollonius
Generalised circles


Circle
Significance in art and symbolism


Orthogonal projections
Spherical tiling
Cartesian coordinates
Equation in three dimensional space
Formulas
Doubling the cube
Uniform colorings and symmetry
Geometric relations
In theology
Other dimensions
Related polyhedra
Cubical graph


Cube
Equation in three dimensional space


Cube
Related polyhedra
In uniform honeycombs and polychora


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


Regular icosahedra
Stellated icosahedra
Pyritohedral symmetry
Other icosahedra


Convex regular icosahedron
Great icosahedron


Icosahedron
Regular icosahedra
Convex regular icosahedron


Icosahedron
Regular icosahedra
Great icosahedron


Cartesian coordinates
Jessen's icosahedron
Cuboctahedron


Icosahedron
Pyritohedral symmetry
Cartesian coordinates


Icosahedron
Pyritohedral symmetry
Jessen's icosahedron


Icosahedron
Pyritohedral symmetry
Cuboctahedron


Rhombic icosahedron
Pyramid and prism symmetries
Johnson solids


Icosahedron
Other icosahedra
Rhombic icosahedron


Icosahedron
Other icosahedra
Pyramid and prism symmetries


Icosahedron
Other icosahedra
Johnson solids


Conic section analysis of the hyperbolic appearance of circles

Hyperbolas as plane sections of quadrics


Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form

As an affine image of the unit hyperbola x² − y² = 1

As an affine image of the hyperbola y = 1/x


Hyperbola
Definitions
As locus of points


Hyperbola
Definitions
Hyperbola with equation y = A/x


Hyperbola
Definitions
By the directrix property


Hyperbola
Definitions
As plane section of a cone


Hyperbola
Definitions
Pin and string construction


Hyperbola
Definitions
Steiner generation of a hyperbola


Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form


As an affine image of the unit hyperbola x² − y² = 1


As an affine image of the hyperbola y = 1/x

Tangent construction
Point construction
Tangent–asymptotes triangle


Hyperbola
Definitions
Reciprocation of a circle


Hyperbola
Definitions
Quadratic equation


Parametric representation with hyperbolic sine/cosine


Hyperbola
In Cartesian coordinates
Equation


Hyperbola
In Cartesian coordinates
Equation
Eccentricity


Hyperbola
In Cartesian coordinates
Asymptotes


Hyperbola
In Cartesian coordinates
Semi-latus rectum


Hyperbola
In Cartesian coordinates
Tangent


Hyperbola
In Cartesian coordinates
Rectangular hyperbola


Hyperbola
In Cartesian coordinates
Conjugate hyperbola


The tangent bisects the angle between the lines to the foci


Hyperbola
Properties
Midpoints of parallel chords


Hyperbola
Properties
Orthogonal tangents – orthoptic


Hyperbola
Properties
Pole-polar relation for a hyperbola


Conic section analysis of the hyperbolic appearance of circles


Sundials
Multilateration
Path followed by a particle
Korteweg–de Vries equation
Angle trisection
Efficient portfolio frontier
Biochemistry


Hyperbola
Applications
Path followed by a particle


Hyperbola
Applications
Korteweg–de Vries equation


Hyperbola
Applications
Efficient portfolio frontier


Hyperbolas as plane sections of quadrics


Regular tetrahedron
Irregular tetrahedra
General properties
Integer tetrahedra
Related polyhedra and compounds
Applications
Tetrahedral graph


Orthogonal projections of the regular tetrahedron


Tetrahedron
Regular tetrahedron
Coordinates for a regular tetrahedron


Tetrahedron
Regular tetrahedron
Angles and distances


Tetrahedron
Regular tetrahedron
Isometries of the regular tetrahedron


Tetrahedron
Regular tetrahedron
Cross section of regular tetrahedron


Tetrahedron
Regular tetrahedron
Spherical tiling


Tetrahedron
Regular tetrahedron
Helical stacking


Trirectangular tetrahedron
Disphenoid
Orthoschemes
Space-filling tetrahedra
Fundamental domains
Isometries of irregular tetrahedra


Tetrahedron
Irregular tetrahedra
Trirectangular tetrahedron


Tetrahedron
Irregular tetrahedra
Disphenoid


Tetrahedron
Irregular tetrahedra
Orthoschemes


Tetrahedron
Irregular tetrahedra
Space-filling tetrahedra


Tetrahedron
Irregular tetrahedra
Fundamental domains


Tetrahedron
Irregular tetrahedra
Isometries of irregular tetrahedra


Properties analogous to those of a triangle

A law of sines for tetrahedra and the space of all shapes of tetrahedra


Heron-type formula for the volume of a tetrahedron


Tetrahedron
General properties
Volume
Volume divider


Tetrahedron
General properties
Volume
Non-Euclidean volume


Tetrahedron
General properties
Distance between the edges


Properties analogous to those of a triangle


Tetrahedron
General properties
Geometric relations


A law of sines for tetrahedra and the space of all shapes of tetrahedra


Tetrahedron
General properties
Law of cosines for tetrahedra


Tetrahedron
General properties
Interior point


Tetrahedron
General properties
Circumradius


Tetrahedron
General properties
Circumcenter


Tetrahedron
Related polyhedra and compounds


Numerical analysis
Structural engineering
Aviation
Chemistry
Electricity and electronics
Color space
Games
Geology
Popular culture


Tetrahedron
Applications
Numerical analysis


Tetrahedron
Applications
Structural engineering


Tetrahedron
Applications
Electricity and electronics


Tetrahedron
Applications
Popular culture


Early life and education
Early political career
Premiership
Presidency
Ideology and public image
Personal life
Electoral history
Honours and accolades
Bibliography


Recep Tayyip Erdoğan
Early life and education


Recep Tayyip Erdoğan
Early life and education
Education


Recep Tayyip Erdoğan
Early political career

Mayor of Istanbul (1994–1998)
Imprisonment
Justice and Development Party


Recep Tayyip Erdoğan
Early political career
Mayor of Istanbul (1994–1998)


Recep Tayyip Erdoğan
Early political career
Imprisonment


Recep Tayyip Erdoğan
Early political career
Justice and Development Party


General elections
Referendums
Domestic policy
Foreign policy
Protests


Recep Tayyip Erdoğan
Premiership
General elections


Recep Tayyip Erdoğan
Premiership
Referendums


Recep Tayyip Erdoğan
Premiership
Domestic policy

Kurdish issue
Armenian genocide
Human rights
Economy
Education
Infrastructure
Justice
Civil–military relations
Health care


Recep Tayyip Erdoğan
Premiership
Domestic policy
Kurdish issue


Recep Tayyip Erdoğan
Premiership
Domestic policy
Armenian genocide


Recep Tayyip Erdoğan
Premiership
Domestic policy
Human rights


Recep Tayyip Erdoğan
Premiership
Domestic policy
Economy


Recep Tayyip Erdoğan
Premiership
Domestic policy
Education


Recep Tayyip Erdoğan
Premiership
Domestic policy
Infrastructure


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Premiership
Domestic policy
Justice


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Premiership
Domestic policy
Civil–military relations


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Premiership
Domestic policy
Health care


Recep Tayyip Erdoğan
Premiership
Foreign policy

European Union
Greece and Cyprus dispute
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Iraq
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Syria
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Recep Tayyip Erdoğan
Premiership
Foreign policy
European Union


Recep Tayyip Erdoğan
Premiership
Foreign policy
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Premiership
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Premiership
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Russia


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Premiership
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Premiership
Protests


Presidential elections
Referendum
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Events


Recep Tayyip Erdoğan
Presidency
Presidential elections


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Presidency
Referendum


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Presidency
Referendum
Local elections


Recep Tayyip Erdoğan
Presidency
Domestic policy

Presidential palace
The media
State of emergency and purges
Economic policy


Recep Tayyip Erdoğan
Presidency
Domestic policy
Presidential palace


Recep Tayyip Erdoğan
Presidency
Domestic policy
The media


Recep Tayyip Erdoğan
Presidency
Domestic policy
State of emergency and purges


Recep Tayyip Erdoğan
Presidency
Domestic policy
Economic policy


Recep Tayyip Erdoğan
Presidency
Foreign policy

Europe
Greece
The Balkans
United Kingdom
Israel
Syrian Civil War
China
Japan
Qatar blockade
Myanmar
United States
Venezuela
Ukraine and Russian invasion of Ukraine


Recep Tayyip Erdoğan
Presidency
Foreign policy
Europe


Recep Tayyip Erdoğan
Presidency
Foreign policy
Greece


Recep Tayyip Erdoğan
Presidency
Foreign policy
The Balkans


Recep Tayyip Erdoğan
Presidency
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United Kingdom


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Presidency
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Israel


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Presidency
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Syrian Civil War


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Presidency
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Japan


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Presidency
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Myanmar


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Presidency
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United States


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Presidency
Foreign policy
Venezuela


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Presidency
Foreign policy
Ukraine and Russian invasion of Ukraine


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Presidency
Events
Coup d'état attempt


Recep Tayyip Erdoğan
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Events
2023 earthquake


Recep Tayyip Erdoğan
Ideology and public image

Ziya Gökalp
Ottomanism
Authoritarianism
Erdoğanism
Views on minorities


Recep Tayyip Erdoğan
Ideology and public image
Ziya Gökalp


Recep Tayyip Erdoğan
Ideology and public image
Ottomanism


Recep Tayyip Erdoğan
Ideology and public image
Authoritarianism


Recep Tayyip Erdoğan
Ideology and public image
Authoritarianism
Suppression of dissent


Recep Tayyip Erdoğan
Ideology and public image
Erdoğanism


Recep Tayyip Erdoğan
Ideology and public image
Views on minorities


Recep Tayyip Erdoğan
Ideology and public image
Views on minorities
LGBT


Recep Tayyip Erdoğan
Ideology and public image
Views on minorities
Jews


Recep Tayyip Erdoğan
Honours and accolades


Recep Tayyip Erdoğan
Honours and accolades
Foreign honours


Recep Tayyip Erdoğan
Honours and accolades
Foreign honours
Supranational


Recep Tayyip Erdoğan
Honours and accolades
Other awards


Background
Prelude
Invasion
Reactions
Foreign involvement
Casualties
Impacts
Peace efforts


Ukrainian unrest, Russian activity in Crimea and Donbas


Russian invasion of Ukraine
Prelude
Putin's invasion  announcement


Initial invasion of Ukraine (24 February – 7 April)

Southeastern front (8 April – 5 September)

Russian annexations and Ukrainian counterattacks (6 September – 11 November 2022)

Winter stalemate, attrition campaign and military surge (12 November 2022 – present)


Initial invasion of Ukraine (24 February – 7 April)

Kyiv and northern front
North-eastern front
Southern front
Eastern front


Southeastern front (8 April – 5 September)

Fall of Mariupol
Fall of Sievierodonetsk and Lysychansk
Kharkiv front
Kherson-Mykolaiv front
Zaporizhzhia front


Russian annexations and Ukrainian counterattacks (6 September – 11 November 2022)

Russian annexation of Donetsk, Kherson, Luhansk and Zaporizhzhia oblasts


Russian annexation of Donetsk, Kherson, Luhansk and Zaporizhzhia oblasts


Winter stalemate, attrition campaign and military surge (12 November 2022 – present)

Battle of Bakhmut
Southern front
Diplomatic developments
Continued infrastructure attacks


Russian invasion of Ukraine
Invasion
Events in Crimea


Russian invasion of Ukraine
Invasion
Missile attacks and aerial warfare


Russian invasion of Ukraine
Invasion
Naval blockade and engagements


Russian invasion of Ukraine
Invasion
Command


Russian invasion of Ukraine
Invasion
Nuclear threats


Russian invasion of Ukraine
Invasion
Ukrainian resistance


Russian invasion of Ukraine
Foreign involvement


Russian invasion of Ukraine
Casualties
Prisoners of war


Humanitarian impact
Refugee crisis
Environmental impact


Russian invasion of Ukraine
Impacts
Humanitarian impact


Russian invasion of Ukraine
Impacts
Refugee crisis


Russian invasion of Ukraine
Impacts
Environmental impact


Russian invasion of Ukraine
Peace efforts


Definition Bi-interpretability Example References


Interpretation (model theory)
Definition


Interpretation (model theory)
Bi-interpretability


Interpretation (model theory)
References


Elementary substructures and elementary extensions


Elementary equivalence
Elementarily equivalent structures


Elementary equivalence
Tarski–Vaught test


Elementary equivalence
Elementary embeddings


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


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


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


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


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


Further terminology
Measurements and equations
Projective geometry
Generalizations


Volume
Center of mass
Right circular cone
Elliptic cone


Cone
Measurements and equations
Center of mass


Cone
Measurements and equations
Right circular cone

Volume
Slant height
Surface area
Circular sector
Equation form


Cone
Measurements and equations
Right circular cone
Volume


Cone
Measurements and equations
Right circular cone
Slant height


Cone
Measurements and equations
Right circular cone
Surface area


Cone
Measurements and equations
Right circular cone
Circular sector


Cone
Measurements and equations
Right circular cone
Equation form


Cone
Measurements and equations
Elliptic cone


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


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


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


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


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 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