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Mathematics

| 🗖 nosplit | ↑ parent "Ciro Santilli's Homepage" | words: 17 | descendant words: 19k | descendants: 492

🗖 nosplit | ⇓ toc | ↑ parent "Ciro Santilli's Homepage" | Wikipedia | words: 17 | descendant words: 19k | descendants: 492

The proper precise definition of mathematics can be found at: Section 2. "Formalization of Mathematics".

Table of contents | 17, 19k, 492
- 1. The beauty of mathematics | 313, 748, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 1.1. Classification (mathematics) | 101 | 🔗 link | 🗖 nosplit | ↑ parent "The beauty of mathematics"
  - 1.2. Pathological (mathematics) | 0, 41, 1 | 🔗 link | 🗖 nosplit | ↑ parent "The beauty of mathematics"
    - 1.2.1. Exceptional object | 41 | 🔗 link | 🗖 nosplit | ↑ parent "Pathological"
  - 1.3. Lists of mathematical problems | 13, 293, 3 | 🔗 link | 🗖 nosplit | ↑ parent "The beauty of mathematics"
    - 1.3.1. Hilbert's problems | 22 | 🔗 link | 🗖 nosplit | ↑ parent "Lists of mathematical problems"
    - 1.3.2. Millennium Prize Problems | 244, 258, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lists of mathematical problems"
      - 1.3.2.1. Birch and Swinnerton-Dyer Conjecture | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Millennium Prize Problems"
- 2. Formalization of Mathematics | 457, 2k, 50 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 2.1. QED manifesto | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.2. Formal proof | 15, 219, 11 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.2.1. Mathematical proof | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
    - 2.2.2. Formal system | 0, 101, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
      - 2.2.2.1. Zermelo-Fraenkel set theory (ZF) | 57, 101, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Formal system"
        
        2.2.2.1.1. Set theory | 28 | 🔗 link | 🗖 nosplit | ↑ parent "Zermelo-Fraenkel set theory"
        
        2.2.2.1.2. Metamath | 16 | 🔗 link | 🗖 nosplit | ↑ parent "Zermelo-Fraenkel set theory"
    - 2.2.3. Axiom | 0, 90, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
      - 2.2.3.1. Consistency | 41 | 🔗 link | 🗖 nosplit | ↑ parent "Axiom"
      - 2.2.3.2. Independence (mathematical logic) | 49 | 🔗 link | 🗖 nosplit | ↑ parent "Axiom"
    - 2.2.4. Conjecture | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
    - 2.2.5. Theorem | 0, 13, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
      - 2.2.5.1. Corollary | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Theorem"
  - 2.3. Set (mathematics) | 86, 188, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.3.1. Cardinality | 102 | 🔗 link | 🗖 nosplit | ↑ parent "Set"
  - 2.4. Function (mathematics) | 24, 795, 19 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.4.1. Bijection | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Function"
      - 2.4.1.1. Injective function | 🔗 link | 🗖 nosplit | ↑ parent "Bijection"
      - 2.4.1.2. Surjective function | 🔗 link | 🗖 nosplit | ↑ parent "Bijection"
    - 2.4.2. Cartesian product | 15 | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.3. Direct product | 40 | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.4. Domain of a function | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.5. Addition | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.6. Subtraction | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.7. Multiplication | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.8. Division | 🔗 link | 🗖 nosplit | ↑ parent "Function"
    - 2.4.9. Exponentiation ( $x^{y}$ ) | 0, 716, 8 | 🔗 link | 🗖 nosplit | ↑ parent "Function"
      - 2.4.9.1. Exponentiation functional equation | 49 | 🔗 link | 🗖 nosplit | ↑ parent "Exponentiation"
      - 2.4.9.2. Exponential function ( $e^{x}$ ) | 0, 667, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Exponentiation"
        
        2.4.9.2.1. Exponential function differential equation | 71 | 🔗 link | 🗖 nosplit | ↑ parent "Exponential function"
        
        2.4.9.2.2. Definition of the exponential function | 0, 519, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Exponential function"
        
        2.4.9.2.2.1. Taylor expansion definition of the exponential function | 85 | 🔗 link | 🗖 nosplit | ↑ parent "Definition of the exponential function"
        
        2.4.9.2.2.2. Product definition of the exponential function | 434 | 🔗 link | 🗖 nosplit | ↑ parent "Definition of the exponential function"
        
        2.4.9.2.3. Logarithm | 🔗 link | 🗖 nosplit | ↑ parent "Exponential function"
        
        2.4.9.2.4. Matrix exponential | 77 | 🔗 link | 🗖 nosplit | ↑ parent "Exponential function"
  - 2.5. Function space | 11 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.6. Integer ( $Z$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.7. Rational number ( $Q$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.8. Real number ( $R$ ) | 13, 13, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.8.1. Dedekind cut | 🔗 link | 🗖 nosplit | ↑ parent "Real number"
    - 2.8.2. Cardinality of the continuum | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Real number"
      - 2.8.2.1. Countable set | 🔗 link | 🗖 nosplit | ↑ parent "Cardinality of the continuum"
      - 2.8.2.2. Cantor's diagonal argument | 🔗 link | 🗖 nosplit | ↑ parent "Cardinality of the continuum"
  - 2.9. Complex number | 43, 141, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.9.1. Quaternion | 78, 98, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Complex number"
      - 2.9.1.1. Octonion | 20 | 🔗 link | 🗖 nosplit | ↑ parent "Quaternion"
  - 2.10. Ordered pair | 37 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.11. Existence and uniqueness | 82, 82, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.11.1. Existence | 🔗 link | 🗖 nosplit | ↑ parent "Existence and uniqueness"
    - 2.11.2. Uniqueness | 🔗 link | 🗖 nosplit | ↑ parent "Existence and uniqueness"
- 3. Algebra | 0, 4k, 110 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 3.1. Commutator | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
  - 3.2. Linear algebra | 0, 544, 31 | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
    - 3.2.1. Linear function | 0, 9, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
      - 3.2.1.1. Multilinear function | 0, 9, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Linear function"
        
        3.2.1.1.1. Bilinear map | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Multilinear function"
        
        3.2.1.1.2. Alternating multilinear map | 🔗 link | 🗖 nosplit | ↑ parent "Multilinear function"
    - 3.2.2. Dot product | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
    - 3.2.3. Cross product ( $x \times y$ ) | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
      - 3.2.3.1. Jacobi identity | 🔗 link | 🗖 nosplit | ↑ parent "Cross product"
    - 3.2.4. Levi-Civita symbol ( $ε$ ) | 38 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
    - 3.2.5. Projection (mathematics) | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
    - 3.2.6. Matrix | 0, 493, 18 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
      - 3.2.6.1. Matrix multiplication | 11, 225, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.6.1.1. Matrix multiplication algorithm | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix multiplication"
        
        3.2.6.1.2. Linear map (linear operator) | 196, 210, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix multiplication"
        
        3.2.6.1.2.1. Linear operator | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Linear map"
        
        3.2.6.1.2.2. Self-adjoint operator | 🔗 link | 🗖 nosplit | ↑ parent "Linear map"
      - 3.2.6.2. Eigenvalues and eigenvectors | 0, 218, 10 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.6.2.1. Applications of eigenvalues and eigenvectors | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.6.2.2. Eigenvalue | 9, 130, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.6.2.2.1. Spectrum (functional analysis) | 18, 121, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalue"
        
        3.2.6.2.2.1.1. Continuous spectrum (functional analysis) | 103 | 🔗 link | 🗖 nosplit | ↑ parent "Spectrum"
        
        3.2.6.2.3. Eigenvector | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.6.2.4. Eigenvectors and eigenvalues of the identity matrix | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.6.2.5. Spectral theorem | 0, 63, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.6.2.5.1. Hermitian matrix | 0, 63, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Spectral theorem"
        
        3.2.6.2.5.1.1. Hermitian operator | 63, 63, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Hermitian matrix"
        
        3.2.6.2.5.1.1.1. Riesz representation theorem | 🔗 link | 🗖 nosplit | ↑ parent "Hermitian operator"
      - 3.2.6.3. Orthogonal matrix | 11, 50, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.6.3.1. Unitary matrix | 39 | 🔗 link | 🗖 nosplit | ↑ parent "Orthogonal matrix"
    - 3.2.7. Vector space | 0, 4, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
      - 3.2.7.1. All vector spaces of the same dimension on a given field are isomorphic | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Vector space"
      - 3.2.7.2. Vector (mathematics) | 🔗 link | 🗖 nosplit | ↑ parent "Vector space"
  - 3.3. Group (mathematics) | 0, 3k, 75 | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
    - 3.3.1. Commutative property | 0, 30, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.1.1. Abelian group | 30, 30, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Commutative property"
        
        3.3.1.1.1. Non-commutative | 🔗 link | 🗖 nosplit | ↑ parent "Abelian group"
    - 3.3.2. Symmetry | 31 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
    - 3.3.3. Important mathematical group | 0, 405, 17 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.3.1. Important discrete mathematical group | 0, 405, 16 | 🔗 link | 🗖 nosplit | ↑ parent "Important mathematical group"
        
        3.3.3.1.1. Cyclic group ( $C_{n}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.2. The direct product of two cyclic groups of coprime order is another cyclic group | 72 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.3. Permutation | 0, 284, 9 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.3.1. Cycle notation | 80 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation"
        
        3.3.3.1.3.2. Parity of a permutation | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation"
        
        3.3.3.1.3.2.1. Odd permutation | 🔗 link | 🗖 nosplit | ↑ parent "Parity of a permutation"
        
        3.3.3.1.3.2.2. Even permutation | 🔗 link | 🗖 nosplit | ↑ parent "Parity of a permutation"
        
        3.3.3.1.3.3. Permutation group | 0, 204, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation"
        
        3.3.3.1.3.3.1. Stabilizer (group) | 119 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation group"
        
        3.3.3.1.3.3.2. Symmetric group | 15, 33, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation group"
        
        3.3.3.1.3.3.2.1. All groups are isomorphic to a subgroup of the symmetric group | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Symmetric group"
        
        3.3.3.1.3.3.3. Alternating group ( $A_{n}$ ) | 52 | 🔗 link | 🗖 nosplit | ↑ parent "Permutation group"
        
        3.3.3.1.4. Dihedral group ( $D_{n}$ ) | 35 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.5. Wallpaper group | 3 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.6. Space group | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
        
        3.3.3.1.7. Klein four-group | 3 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical group"
    - 3.3.4. Group isomorphism | 0, 126, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.4.1. Group homomorphism | 95, 126, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Group isomorphism"
        
        3.3.4.1.1. Fundamental theorem on homomorphisms | 31 | 🔗 link | 🗖 nosplit | ↑ parent "Group homomorphism"
        
        3.3.4.1.2. Kernel (algebra) | 🔗 link | 🗖 nosplit | ↑ parent "Group homomorphism"
    - 3.3.5. Generating set of a group | 0, 215, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.5.1. Cayley graph | 61, 211, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Generating set of a group"
        
        3.3.5.1.1. Cycle graph (algebra) | 150 | 🔗 link | 🗖 nosplit | ↑ parent "Cayley graph"
      - 3.3.5.2. Order of an element of a group | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Generating set of a group"
    - 3.3.6. Direct product of groups ( $G \times H$ ) | 0, 441, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.6.1. Product of group subsets | 🔗 link | 🗖 nosplit | ↑ parent "Direct product of groups"
      - 3.3.6.2. Semidirect product ( $N ⋊ H$ ) | 441 | 🔗 link | 🗖 nosplit | ↑ parent "Direct product of groups"
    - 3.3.7. Subgroup | 0, 2k, 33 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.7.1. Subgroup generated by a group | 🔗 link | 🗖 nosplit | ↑ parent "Subgroup"
      - 3.3.7.2. Quotient group | 51, 2k, 31 | 🔗 link | 🗖 nosplit | ↑ parent "Subgroup"
        
        3.3.7.2.1. Subquotient | 64 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.7.2.2. Relationship between the quotient group and direct products | 308 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.7.2.3. Normal subgroup | 149, 1k, 28 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.7.2.3.1. Simple group | 48, 73, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
        
        3.3.7.2.3.1.1. How to show that a group is simple | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Simple group"
        
        3.3.7.2.3.2. Classification of finite groups | 69, 906, 25 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
        
        3.3.7.2.3.2.1. List of finite groups | 4, 13, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
        
        3.3.7.2.3.2.1.1. GroupNames | 9 | 🔗 link | 🗖 nosplit | ↑ parent "List of finite groups"
        
        3.3.7.2.3.2.2. Classification of finite simple groups | 186, 714, 21 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
        
        3.3.7.2.3.2.2.1. Group of Lie type | 67, 67, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.3.7.2.3.2.2.1.1. Chevalley group | 🔗 link | 🗖 nosplit | ↑ parent "Group of Lie type"
        
        3.3.7.2.3.2.2.2. Sporadic group | 8, 447, 16 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.3.7.2.3.2.2.2.1. Mathieu group | 204, 392, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Sporadic group"
        
        3.3.7.2.3.2.2.2.1.1. k-transitive group | 19, 174, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.1.1.1. Classification of k-transitive groups | 22, 155, 6 | 🔗 link | 🗖 nosplit | ↑ parent "k-transitive group"
        
        3.3.7.2.3.2.2.2.1.1.1.1. 2-transitive group | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.3.7.2.3.2.2.2.1.1.1.1.1. Classification of 2-transitive groups | 🔗 link | 🗖 nosplit | ↑ parent "2-transitive group"
        
        3.3.7.2.3.2.2.2.1.1.1.2. Classification of 3-transitive groups | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.3.7.2.3.2.2.2.1.1.1.3. Classification of 4-transitive groups | 19 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.3.7.2.3.2.2.2.1.1.1.4. Classification of 5-transitive groups | 95 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.3.7.2.3.2.2.2.1.1.1.5. Classification of 6-transitive groups | 10 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.3.7.2.3.2.2.2.1.2. Mathieu group $M_{1} 1$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.1.3. Mathieu group $M_{1} 2$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.1.4. Mathieu group $M_{2} 2$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.1.5. Mathieu group $M_{2} 3$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.1.6. Mathieu group $M_{2} 4$ | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.3.7.2.3.2.2.2.2. Monster group | 33, 47, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Sporadic group"
        
        3.3.7.2.3.2.2.2.2.1. Monstrous moonshine | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Monster group"
        
        3.3.7.2.3.2.2.3. Jordan-Holder Theorem | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.3.7.2.3.2.2.4. Composition series | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.3.7.2.3.2.3. Group extension problem | 110 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
    - 3.3.8. Ring (mathematics) | 241, 512, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.8.1. Field (mathematics) | 81, 271, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Ring"
        
        3.3.8.1.1. Finite field () | 0, 77, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Field"
        
        3.3.8.1.1.1. Classification of finite fields | 77 | 🔗 link | 🗖 nosplit | ↑ parent "Finite field"
        
        3.3.8.1.2. Vector field | 0, 113, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Field"
        
        3.3.8.1.2.1. Algebra over a field | 54, 113, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Vector field"
        
        3.3.8.1.2.1.1. Division algebra | 17, 59, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Algebra over a field"
        
        3.3.8.1.2.1.1.1. Frobenius theorem (real division algebras) | 42 | 🔗 link | 🗖 nosplit | ↑ parent "Division algebra"
  - 3.4. Associative property | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
- 4. Applied mathematics | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
- 5. Calculus | 17, 8k, 180 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 5.1. Mathematical analysis | 17 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 5.2. Limit (mathematics) | 46, 250, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.2.1. Convergent series | 🔗 link | 🗖 nosplit | ↑ parent "Limit"
    - 5.2.2. Continuous function | 0, 116, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Limit"
      - 5.2.2.1. Continuous problems are simpler than discrete ones | 92 | 🔗 link | 🗖 nosplit | ↑ parent "Continuous function"
      - 5.2.2.2. Discrete | 24, 24, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Continuous function"
        
        5.2.2.2.1. Discretization | 🔗 link | 🗖 nosplit | ↑ parent "Discrete"
    - 5.2.3. Infinity ( $\infty$ ) | 88 | 🔗 link | 🗖 nosplit | ↑ parent "Limit"
    - 5.2.4. L'Hôpital's rule (limit of a ratio) | 🔗 link | 🗖 nosplit | ↑ parent "Limit"
  - 5.3. Derivative | 0, 144, 11 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.3.1. Chain rule | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
      - 5.3.1.1. Multivariable chain rule | 🔗 link | 🗖 nosplit | ↑ parent "Chain rule"
    - 5.3.2. Differentiable function | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
    - 5.3.3. Maxima and minima | 0, 4, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
      - 5.3.3.1. Lifegard problem | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Maxima and minima"
      - 5.3.3.2. Saddle point | 🔗 link | 🗖 nosplit | ↑ parent "Maxima and minima"
    - 5.3.4. Newton dot notation | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
    - 5.3.5. Partial derivative | 0, 70, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
      - 5.3.5.1. Partial derivative symbol ( $\partial$ ) | 70 | 🔗 link | 🗖 nosplit | ↑ parent "Partial derivative"
    - 5.3.6. Total derivative | 70 | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
    - 5.3.7. Directional derivative | 🔗 link | 🗖 nosplit | ↑ parent "Derivative"
  - 5.4. Riemann integral | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 5.5. Lebesgue integral | 31, 1k, 15 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.5.1. Lebesgue integral vs Riemann integral | 201, 272, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
      - 5.5.1.1. Real world applications of the Lebesgue integral | 71 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral vs Riemann integral"
    - 5.5.2. Lebesgue measurable | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
    - 5.5.3. Lebesgue integral of $L^{p}$ is complete but Riemann isn't | 91, 805, 11 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
      - 5.5.3.1. Riesz-Fischer theorem | 59, 306, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        5.5.3.1.1. $L^{p}$ is complete | 1 | 🔗 link | 🗖 nosplit | ↑ parent "Riesz-Fischer theorem"
        
        5.5.3.1.2. Fourier basis is complete for $L^{2}$ | 62, 246, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Riesz-Fischer theorem"
        
        5.5.3.1.2.1. $L^{p}$ norm sequence convergence does not imply pointwise convergence | 44 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
        
        5.5.3.1.2.2. Carleson's theorem | 140 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
      - 5.5.3.2. Lp space ( $L^{p}$ ) | 30, 408, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        5.5.3.2.1. $L^{1}$ | 🔗 link | 🗖 nosplit | ↑ parent "Lp space"
        
        5.5.3.2.2. $L^{2}$ | 130, 378, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Lp space"
        
        5.5.3.2.2.1. Plancherel theorem | 150, 248, 2 | 🔗 link | 🗖 nosplit | ↑ parent " $L^{2}$ "
        
        5.5.3.2.2.1.1. The Fourier transform is a bijection in $L^{2}$ | 87 | 🔗 link | 🗖 nosplit | ↑ parent "Plancherel theorem"
        
        5.5.3.2.2.1.2. Every Riemann integrable function is Lebesgue integrable | 11 | 🔗 link | 🗖 nosplit | ↑ parent "Plancherel theorem"
  - 5.6. Measure theory | 57 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 5.7. Fourier series | 143, 555, 8 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.7.1. Applications of the Fourier series | 0, 74, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
      - 5.7.1.1. Solving partial differential equations with the Fourier series | 74 | 🔗 link | 🗖 nosplit | ↑ parent "Applications of the Fourier series"
    - 5.7.2. Discrete Fourier transform (DFT) | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
    - 5.7.3. Fourier transform | 186, 321, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
      - 5.7.3.1. Multidimensional Fourier transform | 43 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
      - 5.7.3.2. Fourier inversion theorem | 41 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
      - 5.7.3.3. Laplace transform | 51 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
    - 5.7.4. History of the Fourier series | 17 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
  - 5.8. Topology | 72, 1k, 37 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.8.1. Covering space | 84, 84, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
      - 5.8.1.1. Double cover | 🔗 link | 🗖 nosplit | ↑ parent "Covering space"
    - 5.8.2. Neighbourhood (mathematics) | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 5.8.3. Topological space | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 5.8.4. Manifold | 164, 424, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
      - 5.8.4.1. Atlas (topology) | 25, 25, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
        
        5.8.4.1.1. Coordinate chart | 🔗 link | 🗖 nosplit | ↑ parent "Atlas"
      - 5.8.4.2. Covariant derivative | 39 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
      - 5.8.4.3. Tangent space | 136, 148, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
        
        5.8.4.3.1. Tangent vector to a manifold | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Tangent space"
      - 5.8.4.4. One-form | 48 | 🔗 link | 🗖 nosplit | ↑ parent "Manifold"
    - 5.8.5. Metric (mathematics) | 0, 54, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
      - 5.8.5.1. Metric space | 10, 54, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Metric"
        
        5.8.5.1.1. Norm (mathematics) | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
        
        5.8.5.1.2. Complete metric space | 44 | 🔗 link | 🗖 nosplit | ↑ parent "Metric space"
    - 5.8.6. Compact space | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 5.8.7. Dense set | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 5.8.8. Homotopy | 0, 526, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
      - 5.8.8.1. Generalized Poincaré conjecture | 282, 526, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Homotopy"
        
        5.8.8.1.1. Exotic sphere | 🔗 link | 🗖 nosplit | ↑ parent "Generalized Poincaré conjecture"
        
        5.8.8.1.2. Poincaré conjecture | 🔗 link | 🗖 nosplit | ↑ parent "Generalized Poincaré conjecture"
        
        5.8.8.1.3. Classification of closed surfaces | 215, 244, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Generalized Poincaré conjecture"
        
        5.8.8.1.3.1. Torus | 🔗 link | 🗖 nosplit | ↑ parent "Classification of closed surfaces"
        
        5.8.8.1.3.2. Möbius strip | 🔗 link | 🗖 nosplit | ↑ parent "Classification of closed surfaces"
        
        5.8.8.1.3.3. Klein bottle | 29 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of closed surfaces"
    - 5.8.9. Real coordinate space ( $R^{n}$ ) | 0, 27, 11 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 5.9. Distribution (mathematics) | 182, 265, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.9.1. Dirac delta function | 83, 83, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Distribution"
      - 5.9.1.1. Green's function | 🔗 link | 🗖 nosplit | ↑ parent "Dirac delta function"
      - 5.9.1.2. Heaviside step function | 🔗 link | 🗖 nosplit | ↑ parent "Dirac delta function"
  - 5.10. Complex analysis | 14, 277, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.10.1. Holomorphic function | 69 | 🔗 link | 🗖 nosplit | ↑ parent "Complex analysis"
    - 5.10.2. Analytic continuation | 52, 194, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Complex analysis"
      - 5.10.2.1. Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) | 16 | 🔗 link | 🗖 nosplit | ↑ parent "Analytic continuation"
      - 5.10.2.2. Identity theorem | 92, 126, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Analytic continuation"
        
        5.10.2.2.1. Riemann zeta function | 0, 34, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Identity theorem"
        
        5.10.2.2.1.1. Riemann hypothesis | 34 | 🔗 link | 🗖 nosplit | ↑ parent "Riemann zeta function"
  - 5.11. Hilbert space | 34, 184, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.11.1. Complete basis | 150 | 🔗 link | 🗖 nosplit | ↑ parent "Hilbert space"
  - 5.12. Differential equation | 0, 3k, 67 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 5.13. Series (mathematics) | 0, 0, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 5.13.1. Power series | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Series"
      - 5.13.1.1. Radius of convergence | 🔗 link | 🗖 nosplit | ↑ parent "Power series"
      - 5.13.1.2. Taylor series | 🔗 link | 🗖 nosplit | ↑ parent "Power series"
  - 5.14. Gradient, Divergence, Curl, and Laplacian | 0, 306, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 5.15. Infinitesimal | 24 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
- 6. Discrete mathematics | 0, 0, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 6.1. Graph (discrete mathematics) | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Discrete mathematics"
    - 6.1.1. Edge (graph) | 🔗 link | 🗖 nosplit | ↑ parent "Graph"
    - 6.1.2. Vertex | 🔗 link | 🗖 nosplit | ↑ parent "Graph"
- 7. Functional equation | 10, 23, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 7.1. Cauchy's functional equation | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Functional equation"
- 8. Game theory | 0, 163, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 8.1. Nash equilibrium | 142, 154, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Game theory"
    - 8.1.1. Mixed strategy | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
    - 8.1.2. Prisoner's dilemma | 4 | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
    - 8.1.3. Bimatrix game | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
  - 8.2. Tit for tat | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Game theory"
- 9. Geometry | 0, 4k, 79 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 9.1. n-sphere ( $S^{n}$ ) | 0, 10, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 9.1.1. Antipodal point | 🔗 link | 🗖 nosplit | ↑ parent "n-sphere"
    - 9.1.2. Diameter | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "n-sphere"
      - 9.1.2.1. Radius | 🔗 link | 🗖 nosplit | ↑ parent "Diameter"
    - 9.1.3. Circle ( $S^{1}$ , 1-sphere) | 🔗 link | 🗖 nosplit | ↑ parent "n-sphere"
    - 9.1.4. Sphere ( $S^{2}$ , 2-sphere) | 🔗 link | 🗖 nosplit | ↑ parent "n-sphere"
    - 9.1.5. 3-sphere ( $S^{3}$ ) | 10 | 🔗 link | 🗖 nosplit | ↑ parent "n-sphere"
  - 9.2. Projective geometry | 0, 321, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 9.2.1. Projective plane | 🔗 link | 🗖 nosplit | ↑ parent "Projective geometry"
    - 9.2.2. Real projective space ( $R P^{n}$ ) | 0, 321, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Projective geometry"
      - 9.2.2.1. Real projective line ( $R P^{1}$ ) | 10 | 🔗 link | 🗖 nosplit | ↑ parent "Real projective space"
      - 9.2.2.2. Real projective plane ( $R P^{2}$ ) | 311 | 🔗 link | 🗖 nosplit | ↑ parent "Real projective space"
  - 9.3. Four-dimensional space (4D) | 10, 38, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 9.3.1. Visualizing 4D | 28 | 🔗 link | 🗖 nosplit | ↑ parent "Four-dimensional space"
  - 9.4. Polytope | 37, 472, 23 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 9.4.1. Convex polytope | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
    - 9.4.2. Regular polytope | 5, 390, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 9.4.2.1. Classification of regular polytopes | 148, 385, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Regular polytope"
        
        9.4.2.1.1. Simplex | 48 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of regular polytopes"
        
        9.4.2.1.2. Hypercube | 112, 132, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of regular polytopes"
        
        9.4.2.1.2.1. Hyperrectangle | 20 | 🔗 link | 🗖 nosplit | ↑ parent "Hypercube"
        
        9.4.2.1.3. Cross polytope | 57 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of regular polytopes"
    - 9.4.3. Polygon | 0, 0, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 9.4.3.1. Regular polygon | 0, 0, 6 | 🔗 link | 🗖 nosplit | ↑ parent "Polygon"
        
        9.4.3.1.1. Regular convex polygon | 0, 0, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Regular polygon"
        
        9.4.3.1.1.1. Triangle | 🔗 link | 🗖 nosplit | ↑ parent "Regular convex polygon"
        
        9.4.3.1.1.2. Square | 🔗 link | 🗖 nosplit | ↑ parent "Regular convex polygon"
        
        9.4.3.1.1.3. Pentagon | 🔗 link | 🗖 nosplit | ↑ parent "Regular convex polygon"
        
        9.4.3.1.1.4. Hexagon | 🔗 link | 🗖 nosplit | ↑ parent "Regular convex polygon"
        
        9.4.3.1.1.5. Octagon | 🔗 link | 🗖 nosplit | ↑ parent "Regular convex polygon"
    - 9.4.4. Polyhedron | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 9.4.4.1. Tetrahedron | 🔗 link | 🗖 nosplit | ↑ parent "Polyhedron"
      - 9.4.4.2. Octahedron | 🔗 link | 🗖 nosplit | ↑ parent "Polyhedron"
    - 9.4.5. Regular polyhedron | 0, 45, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 9.4.5.1. Platonic solid | 45 | 🔗 link | 🗖 nosplit | ↑ parent "Regular polyhedron"
    - 9.4.6. 4-polytope | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 9.4.6.1. Regular 4-polytope | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "4-polytope"
        
        9.4.6.1.1. Tesseract | 🔗 link | 🗖 nosplit | ↑ parent "Regular 4-polytope"
  - 9.5. Differential geometry | 12, 3k, 40 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 9.5.1. Lie group | 278, 3k, 39 | 🔗 link | 🗖 nosplit | ↑ parent "Differential geometry"
      - 9.5.1.1. Applications of Lie groups to differential equations | 29, 133, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        9.5.1.1.1. How to use Lie Groups to solve differential equations | 104 | 🔗 link | 🗖 nosplit | ↑ parent "Applications of Lie groups to differential equations"
      - 9.5.1.2. Lie algebra | 262, 611, 8 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        9.5.1.2.1. Lie group-Lie algebra correspondence | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
        
        9.5.1.2.2. Lie algebra of a matrix Lie group | 81 | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
        
        9.5.1.2.3. Lie bracket | 0, 36, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
        
        9.5.1.2.3.1. Lie bracket of a matrix Lie group | 36 | 🔗 link | 🗖 nosplit | ↑ parent "Lie bracket"
        
        9.5.1.2.4. Exponential map | 12, 142, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
        
        9.5.1.2.4.1. Exponential map (Lie theory) | 130 | 🔗 link | 🗖 nosplit | ↑ parent "Exponential map"
        
        9.5.1.2.5. Baker-Campbell-Hausdorff formula (BCH formula) | 83 | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
        
        9.5.1.2.6. Generator of a Lie algebra | 7 | 🔗 link | 🗖 nosplit | ↑ parent "Lie algebra"
      - 9.5.1.3. Continuous symmetry | 18, 318, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        9.5.1.3.1. Local symmetry | 162, 300, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Continuous symmetry"
        
        9.5.1.3.1.1. Local symmetries of the Lagrangian imply conserved currents | 138 | 🔗 link | 🗖 nosplit | ↑ parent "Local symmetry"
      - 9.5.1.4. Important Lie groups | 0, 1k, 21 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        9.5.1.4.1. General linear group ( $G L (n)$ ) | 44 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        9.5.1.4.2. Special linear group ( $S L (n)$ ) | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        9.5.1.4.3. Orthogonal group ( $O (n)$ ) | 159, 199, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        9.5.1.4.3.1. Special orthogonal group ( $S O (n)$ ) | 22, 40, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Orthogonal group"
        
        9.5.1.4.3.1.1. 3D rotation group ( $S O (3)$ ) | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Special orthogonal group"
        
        9.5.1.4.4. Unitary group ( $U (n)$ ) | 82, 189, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        9.5.1.4.4.1. Unitary group of degree 1 ( $U (1)$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        9.5.1.4.4.2. Unitary group of degree 2 ( $U (2)$ ) | 5 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        9.5.1.4.4.3. Unit circle | 19 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        9.5.1.4.4.4. Special unitary group ( $S U (n)$ ) | 49, 83, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        9.5.1.4.4.4.1. Special unitary of degree 2 ( $S U (2)$ ) | 34 | 🔗 link | 🗖 nosplit | ↑ parent "Special unitary group"
        
        9.5.1.4.5. Poincaré group | 192, 905, 9 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        9.5.1.4.5.1. Galilean transformation | 0, 610, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Poincaré group"
        
        9.5.1.4.5.1.1. Translation (geometry) | 13, 238, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Galilean transformation"
        
        9.5.1.4.5.1.1.1. Translation group | 19, 225, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Translation"
        
        9.5.1.4.5.1.1.1.1. The derivative is the generator of the translation group | 206 | 🔗 link | 🗖 nosplit | ↑ parent "Translation group"
        
        9.5.1.4.5.1.2. Galilean invariance | 285, 372, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Galilean transformation"
        
        9.5.1.4.5.1.2.1. Covariance | 87 | 🔗 link | 🗖 nosplit | ↑ parent "Galilean invariance"
        
        9.5.1.4.5.2. Lorentz group ( $S O (1, 3)$ ) | 61, 103, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Poincaré group"
        
        9.5.1.4.5.2.1. Lorentz boost | 24 | 🔗 link | 🗖 nosplit | ↑ parent "Lorentz group"
        
        9.5.1.4.5.2.2. Indefinite orthogonal group ( $S O (m, n)$ ) | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Lorentz group"
      - 9.5.1.5. Representation theory | 65 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
      - 9.5.1.6. Lie group bibliography | 0, 26, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        9.5.1.6.1. Lie Groups, Physics, and Geometry by Robert Gilmore (2008) | 26 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group bibliography"
- 10. Mathematician | 117, 792, 8 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 10.1. High flying bird vs gophers (Birds and frogs by Freeman Dyson (2009)) | 625 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematician"
  - 10.2. Mathematics Genealogy Project | 🔗 link | 🗖 nosplit | ↑ parent "Mathematician"
  - 10.3. List of mathematicians | 0, 50, 5 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematician"
    - 10.3.1. Évariste Galois | 🔗 link | 🗖 nosplit | ↑ parent "List of mathematicians"
    - 10.3.2. James Harris Simons | 50 | 🔗 link | 🗖 nosplit | ↑ parent "List of mathematicians"
    - 10.3.3. Joseph Fourier | 🔗 link | 🗖 nosplit | ↑ parent "List of mathematicians"
    - 10.3.4. Paul Erdos | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "List of mathematicians"
      - 10.3.4.1. Erdos number | 🔗 link | 🗖 nosplit | ↑ parent "Paul Erdos"
- 11. Mathematical notation | 0, 25, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 11.1. Mathematical symbol that looks like a Greek letter but isn't | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematical notation"
- 12. Number theory | 0, 94, 17 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 12.1. Arithmetic | 50 | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
  - 12.2. Modular arithmetic | 0, 14, 4 | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
  - 12.3. Lowest common denominator (LCD) | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
  - 12.4. Prime number | 0, 30, 9 | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
    - 12.4.1. Goldbach's conjecture | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 12.4.2. Prime number theorem | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 12.4.3. Prime power | 16 | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 12.4.4. Primality test | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 12.4.5. Greatest common divisor (GCD) | 0, 14, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
      - 12.4.5.1. Coprime | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Greatest common divisor"
      - 12.4.5.2. Euclidean algorithm | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Greatest common divisor"
        
        12.4.5.2.1. Extended Euclidean algorithm | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean algorithm"
    - 12.4.6. Least common multiple (lcm) | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
- 13. Numerical analysis | 97, 171, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 13.1. Perturbation theory | 74 | 🔗 link | 🗖 nosplit | ↑ parent "Numerical analysis"
- 14. Polynomial | 0, 165, 10 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 14.1. Algebraic equation | 0, 14, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
    - 14.1.1. Algebraic number | 0, 14, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Algebraic equation"
      - 14.1.1.1. Transcendental number | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Algebraic number"
  - 14.2. Diophantine equation | 12, 48, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
    - 14.2.1. Undecidable Diophantine equation problems | 36 | 🔗 link | 🗖 nosplit | ↑ parent "Diophantine equation"
  - 14.3. Named small order polynomial | 0, 18, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
    - 14.3.1. Linear polynomial | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Named small order polynomial"
  - 14.4. Galois theory | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
  - 14.5. Multivariate polynomial | 60 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
  - 14.6. Polynomial ring | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
- 15. Probability | 0, 0, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 15.1. Expectation value | 🔗 link | 🗖 nosplit | ↑ parent "Probability"
  - 15.2. Standard deviation | 🔗 link | 🗖 nosplit | ↑ parent "Probability"
- 16. Mathematics bibliography | 37, 150, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 16.1. Mathematics YouTube channels | 0, 113, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics bibliography"
    - 16.1.1. 3Blue1Brown | 113 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics YouTube channels"

Ancestors

Ciro Santilli's Homepage | 262, 197k, 3k

Incoming links

Art | 97, 20k, 331
The best articles by Ciro Santilli | 2k
Ciro Santilli's bad old event memory | 663
Classification | 101
Computer science | 260, 3k, 77
Existence and uniqueness | 82, 82, 2
Fields Medal | 17
High budget movies are shit | 131
High flying bird vs gophers | 625
Infinity | 88
Magic: The Gathering | 2k, 2k, 4
Mathematics illustration software | 47
Minimal working example | 64
Molecular Sciences Course of the University of São Paulo | 264
Numerical analysis | 97, 171, 1
Physics | 276, 42k, 744
Quantum electrodynamics | 683, 3k, 29
Lecture 2 | 114
Settheory.net | 69
There is value of tutorials written by early pioneers of the field | 123
Video game | 510, 3k, 40
Website front-end for a mathematical formal proof system | 752