Ciro Santilli
OurBigBook.com
$£
Sponsor
中国
独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
Area of mathematics
Home
Mathematics
OurBigBook.com
words: 26k
articles: 966
Table of contents
26k
966
Formalization of mathematics
Area of mathematics
3k
168
Proof assistant
Formalization of mathematics
206
6
QED manifesto
Proof assistant
Web-based proof assistant
Proof assistant
1
The Math Genome Project
(2023-)
Web-based proof assistant
List of proof assistants
Proof assistant
2
Lean
(proof assistant)
List of proof assistants
Coq
(software)
List of proof assistants
Formal proof
Formalization of mathematics
241
20
Formal proof is useless
Formal proof
Mathematical proof
Formal proof
Formal system
Formal proof
12
5
Zermelo-Fraenkel set theory
(ZF)
Formal system
4
Zermelo-Fraenkel axioms
(ZF)
Zermelo-Fraenkel set theory
1
Zermelo-Fraenkel axioms with the axiom of choice
Zermelo-Fraenkel axioms
Set theory
Zermelo-Fraenkel set theory
Metamath
Zermelo-Fraenkel set theory
Axiom
Formal proof
2
Consistency
Axiom
Independence
(mathematical logic)
Axiom
Open problem in mathematics
Formal proof
90
6
Conjecture
Open problem in mathematics
57
5
Famous conjecture
Conjecture
30
4
Collatz conjecture
(1937-)
Famous conjecture
3
Collatz-like problem
Collatz conjecture
The Busy Beaver Competition: a historical survey by Pascal Michel
(2013)
Collatz conjecture
Erdős' conjecture on powers of 2
Collatz conjecture
Theorem
Formal proof
1
Corollary
Theorem
Lemma
(mathematics)
Formalization of mathematics
Set
(mathematics)
Formalization of mathematics
2
Union
(set theory,
A
⋃
B
)
Set
Cardinality
Set
Function
(mathematics)
Formalization of mathematics
2k
95
Domain, codomain and image
Function
49
7
Bijection
Domain, codomain and image
2
Injective function
Bijection
Surjective function
Bijection
Domain of a function
Domain, codomain and image
Codomain
Domain, codomain and image
1
Endofunction
Codomain
Image
(mathematics)
Domain, codomain and image
Periodic function
Function
2
Square wave
Periodic function
1
Rectangular wave
Square wave
Function by signature
Function
2k
83
Functional function
Function by signature
1
Convolution
Functional function
Set function
Function by signature
2
Cartesian product
Set function
Direct product
Set function
Numeric function
Function by signature
1k
77
Addition
Numeric function
Subtraction
Numeric function
Multiplication
Numeric function
Division
Numeric function
Exponentiation
(
x
y
)
Numeric function
603
13
nth root
Exponentiation
1
Square root
nth root
Exponentiation functional equation
Exponentiation
Exponential function
(
e
x
)
Exponentiation
501
9
Exponential function differential equation
Exponential function
Definition of the exponential function
Exponential function
2
Taylor expansion definition of the exponential function
Definition of the exponential function
Product definition of the exponential function
Definition of the exponential function
Gaussian function
(
e
−
x
2
)
Exponential function
Logarithm
Exponential function
Matrix exponential
Exponential function
71
2
Logarithm of a matrix
Matrix exponential
1
Existence of the matrix logarithm
Logarithm of a matrix
Polynomial
Numeric function
770
56
Degree of a polynomial
Polynomial
Algebraic equation
Polynomial
139
15
Named algebraic equation
Algebraic equation
7
Quadratic equation
Named algebraic equation
2
Quadratic equation modulo n
Quadratic equation
Quadratic formula
Quadratic equation
Cubic equation
Named algebraic equation
Quartic equation
Named algebraic equation
Quintic equation
Named algebraic equation
1
Abel-Ruffini theorem
Quintic equation
Algebraic equation over a field
Algebraic equation
Algebraic equation modulo n
Algebraic equation
Algebraic number
(Root of a polynomial)
Algebraic equation
23
4
Algebraic number field
(
Q
)
Algebraic number
Algebraic function
(Root of a polynomial)
Algebraic number
Transcendental number
Algebraic number
1
Transcendental number conjecture
Transcendental number
Diophantine equation
Polynomial
165
28
Pythagorean triple
(
a
2
+
b
2
=
c
2
)
Diophantine equation
7
7
Euclid's formula
Pythagorean triple
2
There are infinitely many Pythagorean triples
Euclid's formula
Euclid's formula generates all Pythagorean triples
Euclid's formula
Classification of Pythagorean triples
Pythagorean triple
Taxicab number
Pythagorean triple
Fermat's last theorem
Pythagorean triple
1
Andrew Wiles
Fermat's last theorem
Hilbert's tenth problem
(1970)
Diophantine equation
38
7
Hilbert's tenth problem variant
Hilbert's tenth problem
12
6
Decidability of Hilbert's tenth problem in modular arithmetic
Hilbert's tenth problem variant
Decidability of Hilbert's tenth problem of a given degree and number of variables
Hilbert's tenth problem variant
3
Quadratic Diophantine equation
Decidability of Hilbert's tenth problem of a given degree and number of variables
Hilbert's tenth problem is decidable for quadratic equations
Decidability of Hilbert's tenth problem of a given degree and number of variables
Undecidable Diophantine equation example
Decidability of Hilbert's tenth problem of a given degree and number of variables
Hilbert's tenth problem over other rings
Hilbert's tenth problem variant
Additive number theory
Diophantine equation
114
11
Additive basis
Additive number theory
114
10
Additive basis theorem
Additive basis
98
9
Waring's problem
(Every number is a sum of
g
positive numbers to the power of
k
)
Additive basis theorem
73
8
Waring's problem for squares
Waring's problem
3
Lagrange's four-square theorem
(Every natural number is a sum of four squares, 1770)
Waring's problem for squares
Legendre's three-square theorem
(iff not of form
4
a
(
8
b
+
7
)
, 1770)
Waring's problem for squares
Sum of two squares theorem
Waring's problem for squares
Waring problem variant
Waring's problem
51
3
Waring problem with negative numbers allowed
Waring problem variant
1
Sum of three cubes
(Every number has infinitely many representations as the sum of three cubes)
Waring problem with negative numbers allowed
Waring-Goldbach problem
Waring problem variant
Named small order polynomial
Polynomial
1
Linear polynomial
Named small order polynomial
Galois theory
Polynomial
Irreducible polynomial
Polynomial
Multivariate polynomial
Polynomial
Domain of a polynomial
Polynomial
3
Polynomial over a field
(
F
i
e
l
d
[
X
]
)
Domain of a polynomial
Polynomial over a ring
Domain of a polynomial
Polynomial over a commutative ring
Domain of a polynomial
Polynomial ring
Polynomial
Step function
Numeric function
1
Heavyside step function
Step function
Function space
Formalization of mathematics
Number
Formalization of mathematics
57
19
Scientific notation
Number
1
E notation
Scientific notation
Natural number
(
N
, 0, 1, 2, ...)
Number
Integer
(
Z
, ..., -2, -1, 0, 1, 2, ...)
Number
Rational number
(
Q
,
2
1
,
3
4
)
Number
47
7
Fraction
Rational number
10
6
Reduced fraction
Fraction
3
Farey sequence
(
F
n
)
Reduced fraction
1
Farey sunburst
Farey sequence
Number of elements in a Farey sequence
Reduced fraction
Numerator
(Number on top)
Fraction
Denominator
(Number on bottom)
Fraction
Real number
(
R
)
Number
6
Dedekind cut
Real number
Cardinality of the continuum
Real number
2
Countable set
Cardinality of the continuum
Cantor's diagonal argument
Cardinality of the continuum
Mathematical constant
Real number
1
Pi
(
π
)
Mathematical constant
Complex number
Formalization of mathematics
45
7
Complex conjugate
(
x
)
Complex number
Imaginary number
Complex number
Imaginary unit
(
i
)
Complex number
Cayley-Dickson construction
Complex number
3
Quaternion
(
H
, 4)
Cayley-Dickson construction
Octonion
(8)
Cayley-Dickson construction
Sedenion
(16)
Cayley-Dickson construction
Ordered pair
Formalization of mathematics
Logic
Formalization of mathematics
70
9
Propositional logic
Logic
2
Modus ponens
Propositional logic
If and only if
Propositional logic
First-order logic
Logic
63
5
Existential quantification
(
∃
)
First-order logic
3
Existence and uniqueness
Existential quantification
2
Existence
Existence and uniqueness
Uniqueness
Existence and uniqueness
Universal quantification
(
∀
)
First-order logic
Algebra
Area of mathematics
8k
263
Abstract algebra
Algebra
Algebraic structure
Algebra
90
7
Commutator
Algebraic structure
Identity element
Algebraic structure
2
Inverse element
Identity element
1
Invertible
Inverse element
Order
(algebra)
Algebraic structure
2
Degree
(algebra)
Order
Finite algebraic structure
Order
Linear algebra
Algebra
3k
127
Linear function
Linear algebra
Linear map
(linear operator)
Linear algebra
1k
27
Form
(mathematics)
Linear map
Linear form
Linear map
174
3
Matrix representation of a linear form
Linear form
Dual space
(
V
∗
)
Linear form
1
Dual vector
(
e
i
)
Dual space
Linear operator
Linear map
1
Adjoint operator
(
A
†
)
Linear operator
Self-adjoint operator
Linear map
Multilinear map
Linear map
570
18
Bilinear map
Multilinear map
Bilinear form
(
B
(
x
,
y
)
)
Multilinear map
157
2
Matrix representation of a bilinear form
Bilinear form
1
Effect of a change of basis on the matrix of a bilinear form
(
B
2
=
C
T
B
C
)
Matrix representation of a bilinear form
Multilinear form
Multilinear map
Symmetric bilinear map
Multilinear map
80
9
Symmetric bilinear form
Symmetric bilinear map
1
Matrix representation of a symmetric bilinear form
Symmetric bilinear form
Hermitian form
Symmetric bilinear map
1
Matrix representation of a Hermitian form
Hermitian form
Quadratic form
Symmetric bilinear map
Positive definite symmetric bilinear form
Symmetric bilinear map
1
Matrix representation of a positive definite symmetric bilinear form
Positive definite symmetric bilinear form
Skew-symmetric bilinear map
Symmetric bilinear map
Skew-symmetric bilinear form
Symmetric bilinear map
Symmetric multilinear map
Multilinear map
1
Antisymmetric multilinear map
Symmetric multilinear map
Alternating multilinear map
Multilinear map
Dot product
Linear algebra
3
Orthogonality
Dot product
1
Orthonormality
Orthogonality
Angle
Dot product
Cross product
(
x
×
y
)
Linear algebra
1
Jacobi identity
Cross product
Index picking function
Linear algebra
70
3
Kronecker delta
(
δ
i
j
)
Index picking function
Levi-Civita symbol
(
ε
)
Index picking function
1
Levi-Civita symbol as a tensor
Levi-Civita symbol
Projection
(mathematics)
Linear algebra
Matrix
Linear algebra
984
55
Matrix operation
Matrix
42
6
Determinant
(
d
e
t
)
Matrix operation
Matrix inverse
(
M
−
1
)
Matrix operation
1
Invertible matrix
Matrix inverse
Transpose
(
M
T
)
Matrix operation
2
Transpose of a matrix multiplication
Transpose
Inverse of the transpose
Transpose
Matrix multiplication
Matrix
102
9
System of linear equations
Matrix multiplication
76
4
Application of systems of linear equations
System of linear equations
System of linear equations algorithm
System of linear equations
2
LINPACK benchmarks
System of linear equations algorithm
Conjugate gradient method
System of linear equations algorithm
Application of matrix multiplication
Matrix multiplication
Matrix multiplication algorithm
Matrix multiplication
1
General Matrix Matrix Multipliation
Matrix multiplication algorithm
Matrix decomposition
Matrix multiplication
Eigenvalues and eigenvectors
Matrix
501
19
Applications of eigenvalues and eigenvectors
Eigenvalues and eigenvectors
Characteristic polynomial
Eigenvalues and eigenvectors
Eigenvalue
Eigenvalues and eigenvectors
49
2
Spectrum
(functional analysis)
Eigenvalue
1
Continuous spectrum
(functional analysis)
Spectrum
Eigendecomposition of a matrix
Eigenvalues and eigenvectors
63
7
Eigendecomposition of a real symmetric matrix
Eigendecomposition of a matrix
Sylvester's law of inertia
Eigendecomposition of a matrix
41
5
Congruent matrix
Sylvester's law of inertia
1
Matrix congruence can be seen as the change of basis of a bilinear form
Congruent matrix
Matrix similarity
Sylvester's law of inertia
Metric signature
Sylvester's law of inertia
1
Metric signature matrix
Metric signature
Eigenvector
Eigenvalues and eigenvectors
Eigenvectors and eigenvalues of the identity matrix
Eigenvalues and eigenvectors
Spectral theorem
Eigenvalues and eigenvectors
40
3
Hermitian matrix
(complex analogue of symmetric matrix)
Spectral theorem
2
Hermitian operator
Hermitian matrix
1
Riesz representation theorem
Hermitian operator
Kronecker product
Matrix
Named matrix
Matrix
100
16
Dense and sparse matrices
Named matrix
2
Dense matrix
Dense and sparse matrices
Sparse matrix
Dense and sparse matrices
Diagonal matrix
Named matrix
2
Scalar matrix
(
Z
(
V
)
)
Diagonal matrix
1
Identity matrix
(
I
n
)
Scalar matrix
Square matrix
Named matrix
1
Matrix ring
Square matrix
Orthogonal matrix
Named matrix
1
Unitary matrix
Orthogonal matrix
Triangular matrix
Named matrix
Symmetric matrix
Named matrix
18
4
Definite matrix
Symmetric matrix
1
Positive definite matrix
Definite matrix
Skew-symmetric matrix
Symmetric matrix
1
Skew-symmetric form
Skew-symmetric matrix
Vector space
Linear algebra
226
10
Basis
(linear algebra)
Vector space
175
4
Change of basis
Basis
2
Change of basis matrix
Change of basis
Change of basis between symmetric matrices
Change of basis
Linear independence
Basis
Classification of vector spaces
Vector space
Underlying field of a vector space
Vector space
Tensor product
(
⊗
)
Vector space
Vector
(mathematics)
Vector space
1
Scalar
(mathematics)
Vector
Tensor
Linear algebra
487
17
A linear map is a (1,1) tensor
Tensor
Tensor space
(
T
(
m
,
n
)
)
Tensor
1
Order of a tensor
Tensor space
Einstein notation
Tensor
285
13
Raised and lowered indices
Einstein notation
3
Raised index
Raised and lowered indices
Lowered index
Raised and lowered indices
Raising and lowering indices
Raised and lowered indices
Implicit metric signature in Einstein notation
Einstein notation
Einstein notation for partial derivatives
Einstein notation
88
4
Divergence in Einstein notation
(
∂
i
)
Einstein notation for partial derivatives
Laplacian in Einstein notation
(
∂
i
∂
i
)
Einstein notation for partial derivatives
30
2
D'alembert operator in Einstein notation
(
∂
i
∂
i
)
Laplacian in Einstein notation
1
Klein-Gordon equation in Einstein notation
D'alembert operator in Einstein notation
Covariance and contravariance of vectors
Einstein notation
2
Covariant vector
Covariance and contravariance of vectors
Contravariant vector
Covariance and contravariance of vectors
Linear algebra bibliography
Linear algebra
1
Interactive Linear Algebra by Margalit and Rabinoff
Linear algebra bibliography
Group
Algebra
3k
98
Center
(group theory)
Group
Group axiom
Group
10
3
Commutative property
Group axiom
2
Abelian group
Commutative property
1
Non-commutative
Abelian group
Symmetry
Group
1
Symmetry breaking
Symmetry
Important mathematical group
Group
301
19
Important discrete mathematical group
Important mathematical group
188
18
Cyclic group
(
C
n
)
Important discrete mathematical group
The direct product of two cyclic groups of coprime order is another cyclic group
Important discrete mathematical group
Permutation
Important discrete mathematical group
140
11
Cycle notation
Permutation
Parity of a permutation
Permutation
2
Odd permutation
Parity of a permutation
Even permutation
Parity of a permutation
Permutation group
Permutation
21
6
Stabilizer
(group)
Permutation group
Symmetric group
Permutation group
1
All groups are isomorphic to a subgroup of the symmetric group
Symmetric group
Alternating group
Permutation group
9
2
Alternating group of degree 5
Alternating group
1
The alternating groups of degree 5 or greater are simple
Alternating group of degree 5
Dihedral group
(
D
n
)
Important discrete mathematical group
Wallpaper group
Important discrete mathematical group
Space group
Important discrete mathematical group
Klein four-group
Important discrete mathematical group
Finite group
Group
668
28
Classification of finite groups
Finite group
494
27
List of finite groups
Classification of finite groups
1
GroupNames
List of finite groups
Classification of finite simple groups
Classification of finite groups
356
23
Group of Lie type
Classification of finite simple groups
30
2
Chevalley group
Group of Lie type
1
Chevalley groups
A
n
(
q
)
Chevalley group
Sporadic group
Classification of finite simple groups
125
17
Mathieu group
Sporadic group
93
13
k-transitive group
Mathieu group
82
7
Classification of k-transitive groups
k-transitive group
6
2-transitive group
Classification of k-transitive groups
1
Classification of 2-transitive groups
2-transitive group
Classification of 3-transitive groups
Classification of k-transitive groups
Classification of 4-transitive groups
Classification of k-transitive groups
Classification of 5-transitive groups
Classification of k-transitive groups
Classification of 6-transitive groups
Classification of k-transitive groups
Mathieu group
M
1
1
Mathieu group
Mathieu group
M
1
2
Mathieu group
Mathieu group
M
2
2
Mathieu group
Mathieu group
M
2
3
Mathieu group
Mathieu group
M
2
4
Mathieu group
Janko group
Sporadic group
Monster group
Sporadic group
1
Monstrous moonshine
Monster group
Jordan-Holder Theorem
Classification of finite simple groups
Composition series
Classification of finite simple groups
Group extension problem
Classification of finite groups
Group operation
Group
Group isomorphism
Group
9
4
Isomorphism
Group isomorphism
Group homomorphism
Group isomorphism
2
Fundamental theorem on homomorphisms
Group homomorphism
Kernel
(algebra)
Group homomorphism
Generating set of a group
Group
112
6
Finitely generated group
Generating set of a group
Rank of a group
Generating set of a group
Cayley graph
Generating set of a group
1
Cycle graph
(algebra)
Cayley graph
Cycle of an element of a group
Generating set of a group
1
Order of an element of a group
Cycle of an element of a group
Direct product of groups
(
G
×
H
)
Group
2
Product of group subsets
Direct product of groups
Semidirect product
(
N
⋊
H
)
Direct product of groups
Subgroup
Group
457
7
Subgroup generated by a group
Subgroup
Quotient group
Subgroup
45
5
Subquotient
Quotient group
Relationship between the quotient group and direct products
Quotient group
Normal subgroup
Quotient group
11
2
Simple group
Normal subgroup
1
How to show that a group is simple
Simple group
Ring
(mathematics)
Group
750
17
Commutative ring
Ring
Division ring
Ring
Finite ring
Ring
1
Classification of finite rings
Finite ring
Field
(mathematics)
Ring
506
12
Multiplicative inverse
Field
Distributive property
Field
Finite field
(
G
F
(
n
)
)
Field
4
Classification of finite fields
Finite field
Finite field of non-prime order
Finite field
GF(2)
Finite field
GF(4)
Finite field
Quadratically closed field
Field
Vector field
Field
14
3
Algebra over a field
Vector field
11
2
Division algebra
Algebra over a field
1
Frobenius theorem
(real division algebras)
Division algebra
Associative property
Algebra
Algebraic geometry
Algebra
1k
25
The beauty of alebraic geometry
Algebraic geometry
Algebraic curve
Algebraic geometry
Elliptic curve
Algebraic geometry
1k
22
Elliptic curve group
Elliptic curve
2
Elliptic curve point addition
Elliptic curve group
Elliptic curve point multiplication
Elliptic curve group
Domain of an elliptic curve
Elliptic curve
568
16
Not every
x
belongs to the elliptic curve over a non quadratically closed field
Domain of an elliptic curve
1
Number of elements of an elliptic curve
Not every
x
belongs to the elliptic curve over a non quadratically closed field
Elliptic curve over the real numbers
(
E
(
F
)
)
Domain of an elliptic curve
Elliptic curve over the rational numbers
(
E
(
Q
)
)
Domain of an elliptic curve
193
9
Number of elements of an elliptic curve over the rational numbers
Elliptic curve over the rational numbers
106
3
Mordell's theorem
(1922)
Number of elements of an elliptic curve over the rational numbers
29
2
Rank of an elliptic curve over the rational numbers
(
r
)
Mordell's theorem
1
Largest known ranks of an elliptic curve over the rational numbers
Rank of an elliptic curve over the rational numbers
Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p
Elliptic curve over the rational numbers
Birch and Swinnerton-Dyer conjecture
(1965)
Elliptic curve over the rational numbers
53
3
BSD conjecture bibliography
Birch and Swinnerton-Dyer conjecture
2
Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli
(2023)
BSD conjecture bibliography
Notes on Elliptic Curves (II) by BSD
(1965)
BSD conjecture bibliography
Elliptic curve over a finite field
(
E
(
F
p
)
)
Domain of an elliptic curve
2
Number of elements of an elliptic curve over a finite field
Elliptic curve over a finite field
1
Schoof's algorithm
Number of elements of an elliptic curve over a finite field
Elliptic curve bibliography
Elliptic curve
1
Elliptic curve university course
Elliptic curve bibliography
Calculus
Area of mathematics
6k
248
Mathematical analysis
Calculus
Limit
(mathematics)
Calculus
54
7
Convergent series
Limit
Continuous function
Limit
3
Continuous problems are simpler than discrete ones
Continuous function
Discrete
Continuous function
1
Discretization
Discrete
Infinity
(
∞
)
Limit
L'Hôpital's rule
(limit of a ratio)
Limit
Derivative
Calculus
132
19
Chain rule
Derivative
1
Multivariable chain rule
Chain rule
Differentiable function
Derivative
4
Smoothness
Differentiable function
Infinitely differentiable function
Differentiable function
2
Bump function
Infinitely differentiable function
1
Flat top bump function
Bump function
Maxima and minima
Derivative
3
Lifegard problem
Maxima and minima
Derivative test
Maxima and minima
Saddle point
Maxima and minima
Newton dot notation
Derivative
Partial derivative
Derivative
132
4
Partial derivative notation
Partial derivative
3
Partial derivative symbol
(
∂
)
Partial derivative notation
Partial label partial derivative notation
(
∂
x
F
,
∂
y
F
)
Partial derivative notation
Partial index partial derivative notation
(
∂
0
F
,
∂
1
F
)
Partial derivative notation
Total derivative
Derivative
Directional derivative
Derivative
Integral
Calculus
710
19
Area
Integral
1
Volume
Area
Riemann integral
Integral
Lebesgue integral
Integral
488
15
Lebesgue integral vs Riemann integral
Lebesgue integral
1
Real world applications of the Lebesgue integral
Lebesgue integral vs Riemann integral
Lebesgue measurable
Lebesgue integral
Lebesgue integral of
L
p
is complete but Riemann isn't
Lebesgue integral
385
11
Riesz-Fischer theorem
Lebesgue integral of
L
p
is complete but Riemann isn't
139
4
L
p
is complete
Riesz-Fischer theorem
Fourier basis is complete for
L
2
Riesz-Fischer theorem
2
L
p
norm sequence convergence does not imply pointwise convergence
Fourier basis is complete for
L
2
Carleson's theorem
Fourier basis is complete for
L
2
Lp space
Lebesgue integral of
L
p
is complete but Riemann isn't
135
5
L
1
Lp space
L
2
Lp space
60
3
Plancherel theorem
L
2
2
The Fourier transform is a bijection in
L
2
Plancherel theorem
Every Riemann integrable function is Lebesgue integrable
Plancherel theorem
Measure theory
Calculus
Fourier series
Calculus
350
11
Applications of the Fourier series
Fourier series
1
Solving partial differential equations with the Fourier series
Applications of the Fourier series
Discrete Fourier transform
Fourier series
3
Discrete Fourier transform of a real signal
Discrete Fourier transform
Normalized DFT
Discrete Fourier transform
Fast Fourier transform
Discrete Fourier transform
Fourier transform
Fourier series
3
Multidimensional Fourier transform
Fourier transform
Fourier inversion theorem
Fourier transform
Laplace transform
Fourier transform
History of the Fourier series
Fourier series
Topology
Calculus
1k
64
Covering space
Topology
1
Double cover
Covering space
Neighbourhood
(mathematics)
Topology
Topological space
Topology
Manifold
Topology
5
7
Atlas
(topology)
Manifold
1
Coordinate chart
Atlas
Covariant derivative
Manifold
Differentiable manifold
Manifold
Tangent space
Manifold
1
Tangent vector to a manifold
Tangent space
One-form
Manifold
Metric
(mathematics,
d
(
x
,
y
)
)
Topology
193
10
Metric space
Metric
51
9
Metric space vs normed vector space vs inner product space
Metric space
Complete metric space
Metric space
Normed vector space
Metric space
12
2
Inner product space
Normed vector space
1
Inner product
Inner product space
Norm
(mathematics,
∣
x
∣
)
Metric space
2
Norm induced by an inner product
Norm
Metric induced by a norm
Norm
Pseudometric space
Metric space
Compact space
Topology
Dense set
Topology
Connected space
Topology
3
Connected component
Connected space
Simply connected space
Connected space
1
Loop
(topology)
Simply connected space
Homotopy
Topology
167
7
Generalized Poincaré conjecture
Homotopy
9
6
Exotic sphere
Generalized Poincaré conjecture
Poincaré conjecture
Generalized Poincaré conjecture
Classification of closed surfaces
Generalized Poincaré conjecture
3
Torus
Classification of closed surfaces
Möbius strip
Classification of closed surfaces
Klein bottle
Classification of closed surfaces
Real coordinate space
Topology
199
26
Real line
Real coordinate space
Real plane
Real coordinate space
Real coordinate space of dimension three
Real coordinate space
Real coordinate space of dimension four
Real coordinate space
1
Visualizing 4D
Real coordinate space of dimension four
Dimension
Real coordinate space
2
Infinite dimensional
Dimension
1
Finite dimensional
Infinite dimensional
Complex coordinate space
Real coordinate space
52
3
Complex coordinate space of dimension 2
Complex coordinate space
Complex dot product
Complex coordinate space
1
Norm induced by the complex dot product
Complex dot product
Euclidean space
Real coordinate space
20
13
Euclidean metric signature matrix
Euclidean space
Cartesian coordinate system
Euclidean space
Polar coordinate system
Euclidean space
1
Spherical coordinate system
Polar coordinate system
Pythagorean theorem
Euclidean space
Non-Euclidean geometry
Euclidean space
20
7
Elliptic geometry
Non-Euclidean geometry
20
2
Model of elliptic geometry
Elliptic geometry
1
Projective elliptic geometry
Model of elliptic geometry
Hyperbolic gemoetry
Non-Euclidean geometry
3
Hyperbolic functions
Hyperbolic gemoetry
2
Hyperbolic sine
Hyperbolic functions
Hyperbolic cossine
Hyperbolic functions
Distribution
(mathematics)
Calculus
4
Dirac delta function
Distribution
2
Green's function
Dirac delta function
Heaviside step function
Dirac delta function
Normal distribution
Distribution
Complex analysis
Calculus
107
8
Complex analysis bibliography
Complex analysis
1
Complex Analysis by Juan Carlos Ponce Campuzano
Complex analysis bibliography
Holomorphic function
Complex analysis
Analytic continuation
Complex analysis
23
4
Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016)
Analytic continuation
Identity theorem
Analytic continuation
23
2
Riemann zeta function
Identity theorem
1
Riemann hypothesis
Riemann zeta function
Hilbert space
Calculus
1
Complete basis
Hilbert space
Differential equation
Calculus
2k
86
Euler number
(
e
)
Differential equation
5
3
Natural logarithm
(
l
n
(
n
)
,
l
o
g
e
(
n
)
)
Euler number
2
Logarithmic integral function
(
l
i
(
x
)
=
∫
0
x
l
n
t
d
t
)
Natural logarithm
Euler-Mascheroni constant
Natural logarithm
Linear differential equation
Differential equation
1
Holonomic function
Linear differential equation
Order of a differential equation
Differential equation
Ordinary differential equation
(ODE)
Differential equation
5
Existence and uniqueness of solutions of ordinary differential equations
Ordinary differential equation
2
Peano existence theorem
Existence and uniqueness of solutions of ordinary differential equations
Picard-Lindelöf theorem
Existence and uniqueness of solutions of ordinary differential equations
System of ordinary differential equations
Ordinary differential equation
1
System of linear ordinary differential equations
System of ordinary differential equations
Partial differential equation
Differential equation
2k
57
Analytical method to solve a partial differential equation
Partial differential equation
1
Separation of variables
Analytical method to solve a partial differential equation
Numerical method to solve a partial differential equation
Partial differential equation
3
Variational formulation of a partial differential equation
Numerical method to solve a partial differential equation
1
Weak solution
Variational formulation of a partial differential equation
Finite element method
Numerical method to solve a partial differential equation
Important partial differential equation
Partial differential equation
584
36
Laplace's equation
Important partial differential equation
14
5
Legendre polynomials
Laplace's equation
Poisson's equation
Laplace's equation
1
Uniqueness theorem for Poisson's equation
Poisson's equation
Harmonic function
Laplace's equation
1
Spherical harmonic
Harmonic function
Heat equation
Important partial differential equation
1
Heat equation solution with Fourier series
Heat equation
Wave equation
Important partial differential equation
243
27
Damped wave equation
Wave equation
1
Plucked string model
Damped wave equation
Wave equation boundary condition
Wave equation
Wave equation on string with one vibrating side
Wave equation
Wave equation solver
Wave equation
Wave equation solution with Fourier series
Wave equation
The wave equation can be seen as infinitely many infinitesimal coupled oscillators
Wave equation
Lossy 1D Wave Equation
Wave equation
Wave
Wave equation
3
Wavelength
Wave
Standing wave
Wave
Envelope
(waves)
Wave
Polarization
Wave equation
1
String polarization
Polarization
Diffraction
Wave equation
36
4
Huygens-Fresnel principle
Diffraction
33
3
Kirchhoff's diffraction formula
Huygens-Fresnel principle
2
Fraunhofer diffraction
Kirchhoff's diffraction formula
Fresnel diffraction
Kirchhoff's diffraction formula
Refraction
Wave equation
Resonance
Wave equation
Wave interference
Wave equation
1
Interference pattern
Wave interference
2D wave equation on a circular domain
Wave equation
8
2
Bessel function
2D wave equation on a circular domain
1
Fourier-Bessel series
Bessel function
Helmholtz equation
Wave equation
Existence and uniqueness of solutions of partial differential equations
Partial differential equation
Partial differential equation solver
Partial differential equation
71
6
FreeFem
Partial differential equation solver
7
3
FreeFem examples
FreeFem
2
heat-dirichlet.1d.freefem
FreeFem examples
heat-dirichlet-2d-freefem
FreeFem examples
FEniCS Project
Partial differential equation solver
1
Hans Petter Langtangen
FEniCS Project
System of partial differential equations
Partial differential equation
Classification of second order partial differential equations into elliptic, parabolic and hyperbolic
Partial differential equation
4
Elliptic partial differential equation
Classification of second order partial differential equations into elliptic, parabolic and hyperbolic
Parabolic partial differential equation
Classification of second order partial differential equations into elliptic, parabolic and hyperbolic
Hyperbolic partial differential equation
Classification of second order partial differential equations into elliptic, parabolic and hyperbolic
Which boundary conditions lead to existence and uniqueness of a second order PDE
Classification of second order partial differential equations into elliptic, parabolic and hyperbolic
Phase space
Differential equation
Boundary condition
Differential equation
227
9
Initial condition
Boundary condition
Boundary value problem
Boundary condition
Dirichlet boundary condition
Boundary condition
Neumann boundary condition
Boundary condition
4
Cauchy boundary condition
Neumann boundary condition
Robin boundary condition
Neumann boundary condition
Open boundary condition
Neumann boundary condition
Mixed boundary condition
Neumann boundary condition
Time dependent boundary condition
Boundary condition
Control theory
Differential equation
3
Control engineering
Control theory
Control system
Control theory
Feedback loop
Control theory
Series
(mathematics)
Calculus
36
8
Power series
Series
36
7
Analytic function
Power series
36
4
Sine and cossine
Analytic function
3
Sinusoidal
Sine and cossine
Sine
Sine and cossine
Cosine
Sine and cossine
Radius of convergence
Power series
Taylor series
Power series
Gradient, Divergence, Curl, and Laplacian
Calculus
45
7
Curl
(mathematics,
∇
×
)
Gradient, Divergence, Curl, and Laplacian
Nabla symbol
(
∇
)
Gradient, Divergence, Curl, and Laplacian
1
Del
Nabla symbol
Divergence
(
∇
⋅
,
d
i
v
(
)
)
Gradient, Divergence, Curl, and Laplacian
Gradient
(
∇
)
Gradient, Divergence, Curl, and Laplacian
Laplace operator
(
Δ
,
∇
2
)
Gradient, Divergence, Curl, and Laplacian
1
D'alembert operator
(
□
)
Laplace operator
Infinitesimal
Calculus
Geometry
Area of mathematics
6k
167
Minimum bounding box
Geometry
Bounding box
Geometry
Fractal
Geometry
Point
(geometry)
Geometry
3
Line
(geometry)
Point
Hyperplane
Point
1
Plane
(geometry)
Hyperplane
n-sphere
(
S
n
)
Geometry
6
9
Antipodal point
n-sphere
Diameter
n-sphere
1
Radius
Diameter
Circle
(
S
1
)
n-sphere
2
Squaring the circle
Circle
Tarski's circle-squaring problem
(Cut a circle into square)
Circle
Sphere
(
S
2
)
n-sphere
1
Great circle
Sphere
3-sphere
(
S
3
)
n-sphere
Projective geometry
Geometry
660
12
Projective space
(
P
(
V
)
)
Projective geometry
1
Projective plane
Projective space
Real projective space
(
R
P
n
,
P
(
R
n
+
1
)
)
Projective geometry
559
9
Real projective line
(
R
P
1
,
P
(
R
2
)
)
Real projective space
Real projective plane
(
R
P
2
,
P
(
R
3
)
)
Real projective space
259
7
Synthetic geometry of the real projective plane
Real projective plane
Model of the real projective plane
Real projective plane
2
Lines through origin model of the real projective plane
Model of the real projective plane
Spherical cap model of the real projective plane
Model of the real projective plane
The real projective plane is not simply connected
Real projective plane
Point at infinity
Real projective plane
Homogenous coordinates
Real projective plane
Polytope
Geometry
266
28
Convex polytope
Polytope
Regular polytope
Polytope
150
5
Classification of regular polytopes
Regular polytope
6
4
Simplex
Classification of regular polytopes
Hypercube
Classification of regular polytopes
1
Hyperrectangle
Hypercube
Cross polytope
Classification of regular polytopes
Polygon
Polytope
1
12
Quadrilateral
Polygon
1
Rectangle
Quadrilateral
Parallelogram
Polygon
2
Parallelepiped
Parallelogram
1
Volume of the parallelepiped
Parallelepiped
Regular polygon
Polygon
6
Regular convex polygon
Regular polygon
5
Triangle
Regular convex polygon
Square
Regular convex polygon
Pentagon
Regular convex polygon
Hexagon
Regular convex polygon
Octagon
Regular convex polygon
Polyhedron
Polytope
2
Tetrahedron
Polyhedron
Octahedron
Polyhedron
Regular polyhedron
Polytope
1
Platonic solid
Regular polyhedron
4-polytope
Polytope
2
Regular 4-polytope
4-polytope
1
Tesseract
Regular 4-polytope
Differential geometry
Geometry
5k
101
Lie group
Differential geometry
4k
100
Lie derivative
Lie group
Applications of Lie groups to differential equations
Lie group
Lie algebra
Lie group
194
15
Infinitesimal generator
Lie algebra
Lie group-Lie algebra correspondence
Lie algebra
50
7
Lie algebra exponential covering problem
Lie group-Lie algebra correspondence
2
A single exponential map is not enough to recover a simple Lie group from its algebra
Lie algebra exponential covering problem
The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra
Lie algebra exponential covering problem
Two different Lie groups can have the same Lie algebra
Lie group-Lie algebra correspondence
6
3
Every Lie algebra has a unique single corresponding simply connected Lie group
Two different Lie groups can have the same Lie algebra
1
Universal covering group
Every Lie algebra has a unique single corresponding simply connected Lie group
Every Lie group that has a given Lie algebra is the image of an homomorphism from the universal cover group
Two different Lie groups can have the same Lie algebra
Lie bracket
Lie algebra
Exponential map
Lie algebra
1
Exponential map
(Lie theory)
Exponential map
Baker-Campbell-Hausdorff formula
(BCH formula)
Lie algebra
Generator of a Lie algebra
Lie algebra
Generators of a Lie algebra
Lie algebra
Continuous symmetry
Lie group
97
2
Local symmetry
Continuous symmetry
1
Local symmetries of the Lagrangian imply conserved currents
Local symmetry
Important Lie group
Lie group
3k
67
Matrix Lie group
Important Lie group
142
4
Every closed subgroup of
G
L
(
n
,
C
)
is a Lie group
Matrix Lie group
Lie algebra of a matrix Lie group
Matrix Lie group
2
Lie bracket of a matrix Lie group
Lie algebra of a matrix Lie group
One parameter subgroup
Lie algebra of a matrix Lie group
Classical group
Important Lie group
3
Symplectic group
(
S
p
(
n
,
F
)
)
Classical group
2
Symplectic matrix
Symplectic group
Unitary symplectic group
(
S
p
(
n
)
)
Symplectic group
General linear group
Important Lie group
1
Finite general linear group
(
G
L
(
n
,
F
m
)
)
General linear group
Lie algebra of
G
L
(
n
)
Important Lie group
Special linear group
(
S
L
(
n
)
)
Important Lie group
486
4
Special linear group of dimension 2
(
S
L
(
2
)
)
Special linear group
Lie algebra of
S
L
(
n
)
Special linear group
1
Lie algebra of
S
L
(
2
)
Lie algebra of
S
L
(
n
)
Finite special general linear group
Special linear group
Isometry group
Important Lie group
1
Lie algebra of a isometry group
Isometry group
Orthogonal group
Important Lie group
758
23
Definition of the orthogonal group
Orthogonal group
104
5
The orthogonal group is the group of all matrices that preserve the dot product
Definition of the orthogonal group
1
What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms
The orthogonal group is the group of all matrices that preserve the dot product
The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
Definition of the orthogonal group
1
Elements of the orthogonal group have determinant plus or minus one
The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns
Definition of the orthogonal group
Topology of the orthogonal group
Orthogonal group
2
The orthogonal group is compact
Topology of the orthogonal group
Connected components of the orthogonal group
Topology of the orthogonal group
Lie algebra of
O
(
n
)
Orthogonal group
Special orthogonal group
Orthogonal group
16
3
Lie algebra of
S
O
(
3
)
Special orthogonal group
1
Lie bracket of the rotation group
Lie algebra of
S
O
(
3
)
3D rotation group
Special orthogonal group
Unitary group
Orthogonal group
9
8
Unitary group of degree 1
Unitary group
Unitary group of degree 2
Unitary group
Unit circle
Unitary group
Special unitary group
Unitary group
4
4
Special unitary of degree 2
Special unitary group
4
3
Representations of
S
U
(
2
)
Special unitary of degree 2
2
Lie algebra of
S
U
(
2
)
Representations of
S
U
(
2
)
2D representation of
S
U
(
2
)
Representations of
S
U
(
2
)
Projective linear group
Important Lie group
8
5
Finite projective linear group
Projective linear group
Projective special linear group
Projective linear group
8
3
Finite projective special linear group
Projective special linear group
8
2
P
S
L
(
2
,
p
)
Finite projective special linear group
1
PSL(2,7)
P
S
L
(
2
,
p
)
Poincaré group
Important Lie group
865
17
Galilean transformation
Poincaré group
266
6
Translation
(geometry)
Galilean transformation
121
2
Translation group
Translation
1
The derivative is the generator of the translation group
Translation group
Galilean invariance
Galilean transformation
43
2
Covariance
Galilean invariance
1
Invariant vs covariant
Covariance
Lorentz group
Poincaré group
302
9
Representation theory of the Lorentz group
Lorentz group
8
3
Representation of the Lorentz group
Representation theory of the Lorentz group
2
Lie algebra of the Lorentz group
Representation of the Lorentz group
Spinor
Representation of the Lorentz group
Lorentz boost
Lorentz group
Indefinite orthogonal group
Lorentz group
135
3
Definition of the indefinite orthogonal group
Indefinite orthogonal group
1
All indefinite orthogonal groups of matrices of equal metric signature are isomorphic
Definition of the indefinite orthogonal group
Indefinite special orthogonal group
Indefinite orthogonal group
Representation theory
Lie group
3
Irreducible representation
Representation theory
1
Casimir element
Irreducible representation
Schur's lemma
Representation theory
Simple Lie group
Lie group
1
Classification of simple Lie groups
Simple Lie group
Lie group bibliography
Lie group
4
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011)
Lie group bibliography
Lie Groups, Physics, and Geometry by Robert Gilmore (2008)
Lie group bibliography
Naive Lie theory by John Stillwell (2008)
Lie group bibliography
Lie Algebras In Particle Physics by Howard Georgi (1999)
Lie group bibliography
Tesselation
Geometry
1
5
Aperiodic tiling
Tesselation
Tiling of the plane
Tesselation
1
3
Aperiodic monotile
Tiling of the plane
2
Smith aperiodic monotile
(needs reflections, March 2023)
Aperiodic monotile
Spectre aperiodic monotile
(no reflections, May 2023)
Aperiodic monotile
Number theory
Area of mathematics
716
37
Arithmetic
Number theory
15
Computational number theory
Number theory
Modular arithmetic
Number theory
47
5
Modulo operation
Modular arithmetic
Modular multiplication
Modular arithmetic
1
Modular multiplicative inverse
Modular multiplication
Modular exponentiation
Modular arithmetic
47
1
Computational complexity of modular exponentiation
Modular exponentiation
40
Lowest common denominator
(LCD)
Number theory
Prime number
Number theory
654
27
Type of prime number
Prime number
306
10
Are there infinitely many primes with property X
Type of prime number
36
Mersenne prime
(
2
n
−
1
)
Type of prime number
2
Are there infinitely many Mersenne primes?
Mersenne prime
Lucas-Lehmer primality test
Mersenne prime
Prime k-tuple
(p, p+i, p+j...)
Type of prime number
257
5
Admissible prime k-tuple
Prime k-tuple
188
2
Prime k-tuple conjecture
Admissible prime k-tuple
188
1
Yitang Zhang's theorem
(2013)
Prime k-tuple conjecture
132
Twin prime
Prime k-tuple
69
1
Twin prime conjecture
Twin prime
69
Goldbach's conjecture
(Every even natural number greater than 2 is the sum of two prime numbers)
Prime number
1
Sums of three cubes
Goldbach's conjecture
Prime number theorem
Prime number
224
1
prime-number-theorem
Prime number theorem
224
Prime power
Prime number
9
Primality test
Prime number
28
2
Elliptic curve primality
Primality test
28
AKS primality test
(2002, First polynomial time primality test discovered)
Primality test
Greatest common divisor
(GCD)
Prime number
87
6
Greatest common divisor algorithm
Greatest common divisor
Euclidean algorithm
Greatest common divisor
1
Extended Euclidean algorithm
Euclidean algorithm
Coprime
Greatest common divisor
22
2
Euler's totient function
(
φ
(
n
)
, varphi)
Coprime
15
1
Totient summatory function
(
Φ
(
n
)
, uppercase phi)
Euler's totient function
Least common multiple
(lcm)
Prime number
Numerical analysis
Area of mathematics
221
9
Floating-point arithmetic
Numerical analysis
28
3
Floating-point number
Floating-point arithmetic
IEEE 754
Floating-point arithmetic
28
1
Half-precision floating-point format
(16-bit floating point)
IEEE 754
Numerical computing language
Numerical analysis
81
3
Scilab
Numerical computing language
MATLAB
Numerical computing language
2
1
MATLAB methlab pun
MATLAB
2
Perturbation theory
Numerical analysis
43
Probability
Area of mathematics
21
14
Expectation value
Probability
Stochastic process
Probability
21
4
Markov chain
Stochastic process
21
3
Average number of steps until reaching a state of a Markov chain
Markov chain
4
Average number of steps spent on a node of a Markov chain
Markov chain
4
Absorbing Markov chain
Markov chain
Probability distribution
Probability
Cumulative distribution function
Probability
Standard deviation
Probability
Statistics
Probability
4
Descriptive statistics
Statistics
3
Arithmetic mean
Descriptive statistics
Mode
(statistics)
Descriptive statistics
Variance
Descriptive statistics
Applied mathematics
Area of mathematics
Discrete mathematics
Area of mathematics
445
40
Graph
(discrete mathematics)
Discrete mathematics
445
39
Graph representation
Graph
1
Adjacenty list
Graph representation
Edge
(graph)
Graph
Vertex
Graph
Graph software
Graph
2
6
Graphviz
Graph software
2
5
Graphviz example
Graphviz
2
4
graphviz/hello.dot
Graphviz example
graphviz/quotes.dot
Graphviz example
graphviz/node.dot
Graphviz example
graphviz/quotes-escape.dot
Graphviz example
Type of graph
Graph
443
27
Acyclic graph
Type of graph
Tree
(data structure)
Type of graph
443
20
Tree representation
Tree
177
2
Adjacency list
Tree representation
Nested set model
Tree representation
177
Tree type
Tree
5
Binary tree
Tree type
1
K-ary tree
Binary tree
Ordered and unordered trees
Tree type
2
Ordered tree
Ordered and unordered trees
Unordered tree
Ordered and unordered trees
Tree traversal
Tree
266
10
Depth-first search
Tree traversal
197
8
Pre-order depth-first search
(NLR, Preorder DFS)
Depth-first search
50
1
Iterative pre-order
Pre-order depth-first search
19
In-order depth-first search
(LNR, Inorder DFS)
Depth-first search
128
1
Iterative in-order
In-order depth-first search
57
Post-order depth-first search
(LRN, Postorder DFS)
Depth-first search
19
3
Iterative post-order
Post-order depth-first search
9
2
Iterative post-order with two stacks
Iterative post-order
Iterative post-order with one stack
Iterative post-order
Breadth-first search
Tree traversal
Directed and undirected graphs
Type of graph
3
Directed graph
Directed and undirected graphs
1
Directed acyclic graph
Directed graph
Undirected graph
Directed and undirected graphs
Weighted graph
Type of graph
Game theory
Area of mathematics
289
9
Balance of power
Game theory
Game win predictor
Game theory
1
Elo rating system
Game win predictor
Nash equilibrium
Game theory
266
4
Penalty kick left right Nash equilibrium
Nash equilibrium
118
Mixed strategy
Nash equilibrium
Prisoner's dilemma
Nash equilibrium
Bimatrix game
Nash equilibrium
1
Tit for tat
Game theory
2
Recreational mathematics
Area of mathematics
Ancestors
(2)
Mathematics
Home
Ciro Santilli
OurBigBook.com