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==='''Common Lie Algebras'''===
==='''Common Lie Algebras'''===
* '''𝔰𝔬(3)''' - The '''Lie algebra''' of '''SO(n)''' and would be the '''rate of change''' for Roll, Pitch and Yaw. 𝔰𝔬(n) is equal to 𝔬(n) - Princeton Mathematics (p234)
* '''𝔰𝔬(3)''' - The '''Lie algebra''' of '''SO(n)''' and would be the '''rate of change''' for Roll, Pitch and Yaw. '''𝔰𝔬(n)''' is equal to '''𝔬(n)''' - Princeton Mathematics (p234)
* '''𝔤𝔩ₙ(ℂ)''' is the '''Lie algebra''' for '''GLₙ(ℂ)''', the space of all '''n×n complex matrices'''.
* '''𝔤𝔩ₙ(ℂ)''' is the '''Lie algebra''' for '''GLₙ(ℂ)''', the space of all '''n×n complex matrices'''.
* '''𝔰𝔩ₙ(ℂ)''' - The '''Lie algebra''' of the special linear group '''SLₙ(n)'''. "Is the subspace of all matrices with '''trace zero'''" - Princeton Mathematics (p234)
* '''𝔰𝔩ₙ(ℂ)''' - The '''Lie algebra''' of the special linear group '''SLₙ(n)'''. "Is the subspace of all matrices with '''trace zero'''" - Princeton Mathematics (p234)

Revision as of 12:36, 6 March 2017

Misc

desmos geometry

Read this - About maths, vector spaces, fields, etc... and programming.

Parabola point

Notation and Symbols

unicode.org - UNICODE SUPPORT FOR MATHEMATICS

Operations

Wikipedia: Mathematical operators and symbols in Unicode

Wikipedia: Supplemental Mathematical Operators

  • ⊕ - XOR. Direct Sum.
  • ⊖ - Symmetric difference.
  • ⊗ - Tensor product.
  • ⊘ -
  • ⊙ - Circled dot operator. ⨀ n-ary circled dot operator.
  • ∗ - Astrisk Operator. Also ✱.
  • ∘ - Function composition. Ring operator. (Also ∙ bullet operator)
  • ⋈ - Natural join.
  • ⋆ - Star Operator.
  • ζ(s) - The Riemann zeta function.
  • Arithmetic: − + ± × ⋅ · ÷ ⁄ √ ∛ ∜

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle f(x) = x^2\,}

Template:Math

Calculus

  • ∆ - ∆x change in x. Sometimes symmetric difference.
  • ∇ - Gradient.
  • ∂ - Partial Derivative.
  • ∫ ∬ ∭ ∮ ∯ ∰ - Integrals (double+triple). ∮ ∯ ∰ - Contour, surface and volume integrals. ∱ ∲ ∳ - Clockwise integral, clowise contour integral. [1]

Other

  • ≠ ≈ ≡ ∃ ∄ ∈ ∉ ∌ ⇔ ∀ ∧ ∨ ¬ ≤ ≥ ≮ ≯ ⇒ ∩ ∪ ⋂ ⋃ ⊃ ⊂ ⊇ ⊆ ⊢
  • ◅ ▻ - Normal sub group of. ⟨ ⟩ generator.
  • x′ - x prime.
  • Misc: ∑ ℵ ℶ ω ∞ ∴ ∵ ! ∎ → ↦ ㏒ * ∥ ∦ θ
  • 𝓋⃗, v⃗ - Vector 𝓋 + (U+20D7).
  • ∅ - empty set
  • δ - Kronecker Delta
  • ∏ - Product over (Like ∑). ∐ for coproduct.
  • ⊤ - True
  • ⊥ - False
  • ∖ - Set minus.
  • σ - Selection
  • ↔⇔≡ - If and only if.
  • ⇒ - If A is true then B must be true.
  • ℜ(z), ℑ(z) return the real & imaginary parts of a complex number.
  • {♠, ♦, ♥, ♣}


  • Subscript: ₀₁₂₃₄₅₆₇₈₉ ₊₋₌₍₎ ₐₑₒₓₔₕₖₗₘₙₚₛₜ
  • Superscript: ⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ
  • Blackboard Bold: 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ 𝕒𝕓𝕔𝕕𝕖𝕗𝕘𝕙𝕚𝕛𝕜𝕝𝕞𝕟𝕠𝕡𝕢𝕣𝕤𝕥𝕦𝕧𝕨𝕩𝕪𝕫 𝟘𝟙𝟚𝟛𝟜𝟝𝟞𝟟𝟠𝟡 ℾℽℿℼ⅀ ⅅⅆⅇⅈⅉ


  • wikipedia
  • Mathematical Italic: 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧


  • |𝐴|×|𝐵|

Number Systems

  • ℕ - The set of all natural numbers. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
  • ℤ - The set of all integers. Subtraction is always possible. ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℤ is an infinite ring.
  • ℤₘ - The set of integers modulo m. A finite ring.
  • ℚ - The set of all rational numbers (Quotient). Fractions. Division is always possible. ℚ is a field and an infinite ring.
  • ℝ - The set of all real numbers. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger field. Also an infinite ring.
  • ℝ⁺ - The set of all positive reals.
  • ℂ - The set of all complex numbers. 𝒾²=−1. Is a field and an infinite ring. Is not ordered because there is no way to tell which complex number is > another.
  • ℍ - The set of all Quaternions. Are noncommutative. 𝒾𝒿=𝓀, 𝒿𝒾=−𝓀. Not a field.
  • 𝕆 - Octonions. An 8 dimension number system. Noncommutative and nonassociative. Not a field.
  • 𝔽ₚ - Set of integers modulo a prime p. Addition and multiplication also defined modulo p. A field.
  • 𝕂 - Field of real or complex numbers.
  • 𝔽 - Finite field.
  • ℚ(γ) - γ is the root of the polynomial 𝓍³-𝓍-1. When γ³ appears it is replaced with γ+1. See: 2.2 in Princeton companion of mathematics. AUTOMORPHISM
  • ℙ - Primes.
  • 𝔸ⁿ - Affine Space. A n-dimensional real vector space without origin. ℝⁿ/translations. 2 Objects points and vectors. ~3:00min in
  • ℙⁿ - n-dimensional projective space. ℝⁿ/scaling. If you can rescale one vector to another, they are the same.
  • ℤ/pℤ - The set of integers modulo some prime p. Or just ℤ/p for short.

Algebraic Structures

Properties

  • associative - a(bc) = (ab)c. ∀x,y,z. x∗(y∗z)=(x∗y)∗z
  • commutative - ab = ba. ∀x,y. x∗y=y∗x
  • Identity element: There exists an element e such that for each element x, e ∗ x = x = x ∗ e; formally: ∃e ∀x. e∗x=x=x∗e.
  • Inverse element: It can easily be seen that the identity element is unique. If this unique identity element is denoted by e then for each x, there exists an element i such that x∗i=e=i∗x; formally: ∀x ∃i. x∗i=e=i∗x.
  • homomorphism - A category of function. Preserves the structure. f: X→Y. f(a)f(b)=f(c) and ab=c. A homomorphism of vector spaces is refered to as a linear map.
  • bijection - A bijection is invertable.
  • isomorphism - A homomorphism that is also a bijection. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. f: X→Y, g: Y→X
  • automorphism - Galios Groups related. An isomorphism from a structure to itself.
  • Abelion - An abelian group/operation is commutative.

Structures

Set

  • A collection of elements.

Group

An algebraic structure that is a set with a single binary operation.

An infinite group has an unlimited number of elements. (ℤ, +)

A finite groups has a finite number of elements. (ℤₘ, +) - The integers mod m.

The order of an element is the number of elements in the subgroup it generates. |2| = 3

  • (𝔾, ∗) - Group 𝔾 with operation ∗.
  • Axioms
    • The operation must be closed. - a,b ∈ 𝔾 → a∗b ∈ 𝔾
    • The operation must be associative. - (a∗b)∗c = a∗(b∗c)
    • Operation must have an identity element that has no effect on any other element under operation. ∃e(a∗e = e∗a = a).
    • Every element must have an inverse. An element when combined with the original will return the identity. ∀a∃a⁻¹(a∗a⁻¹=a⁻¹∗a=e)
  • (ℤ, +) - The integers under addition.
    • Closure: a,b ∈ ℤ → a+b ∈ ℤ
    • Associativity: (a+b)+c = a+(b+c)
    • Identity: a+0 = 0+a = a
    • Inverse: a⁻¹ = -a
  • (ℤ, ×) - The integers under multiplication it not a group
    • Closure: a,b ∈ ℤ → a×b ∈ ℤ
    • Associativity: (a×b)×c = a×(b×c)
    • Identity: a×1 = 1×a = a
    • NO Inverse: 2x=1 has no solution in ℤ
  • If the operation is also commutative then it is an Abelian group.
  • Bill Shillito.

Ring

Similar to a field but multiplication doesn't require an inverse.

ℤ, ℚ, ℝ, ℂ, ℤₘ are all unital commutative rings.

𝕄₂(ℝ) the set of all 2×2 real matrices is a non-commutative ring as matrix multiplication is not commutative.

  • (R, +, ∗)
    • (R, +) is an abelian group - addition
      • Closed
      • Associative
      • Identity - Additive identity
      • Inverse - Additive inverse −a
      • Commutative
    • (R, ∗)
      • Closed under ×
      • Operation × must be associative.
    • + and × must be linked by the distributive property. Multiplication distributes over addition.
      • a×(b+c) = a×b+a×c
      • (a+b)×c = a×c+b×c
        • 0×a = a×0 = 0 - Zero times anything is zero
        • a×−b = −a×b = −(a×b) - A positive times a negative is a negative.
        • −a×−b = a×b - A negative times a negative is a positive.
  • × might not be commutative. If it is then it is a commutative ring (not called abelian which is only for groups).
  • If the ring has a multiplicative identity then it is a unital ring. For ℤ, ℚ, ℝ, ℂ, ℤₘ the multiplicative identity is 1. For 𝕄ₘ(ℝ) it is the identity matrix.
  • An element of a unital ring that has a multiplicative inverse is called a unit. Not every element is necessarily a unit. In ℤ only 1 and −1.
  • If every element other than 0 is a unit. It is a division ring. ℚ is a division ring.
  • A zero divisor of a ring is a nonzero element that can be multiplied by some other nonzero element to produce 0. In ℤ₆, 2, 3 and 4 are zero divisors. 2×3=0
  • An element can not be a unit and a zero divisor.
  • An integral domain is a commutative, unital ring with no zero divisors. ℤ, ℚ, ℝ, ℂ are integral domains.
    • ℤₘ is not necessarily an integral domain. ℤₚ is an integral domain if p is a prime number.
    • In an integral domain you can cancel factors from both sides of an equation. 2×x=2×3

Polynomial Ring

A monomial is the product of a number and a non-negative integer power of a variable. eg: 2x³

A polynomial is a finite sum of monomials. a₀+a₁x+a₂x²+a₃x³+…+aₙxⁿ

n is the degree of the polynomial. It's highest power.

Some polynomial rings: ℤ[x], ℚ[x], ℝ[x], ℂ[x], ℤₘ[x]

Field

Like a ring with stricter multiplication axioms.

A field is a division ring where × is commutative, unital and has no zero divisors and every nonzero element is a unit.

A field forms an abelian group under both addition and multiplication.

ℚ, ℝ, ℂ are fields.

ℤₚ is a field if p is prime.

ℤ is not a field.

  • A set with 2 binary operations (+, ×)
  • Both must be commutative and associative
  • Operations must have identity elements (+ is 0 and × is 1)
  • Every element must have an inverse (x = -x, x = 1/x)
  • Must follow the distributive law.
  • Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).

ℝ is the completion of ℚ. It allows for calculus.

ℂ is algebraically closed. Every polynomial equation in ℂ[x] has solutions in ℂ.

But ℂ is not ordered.

Vector Space

    • Like a 2D or 3D plane. Can be built from 'unit vectors' or 'basis'. for example (1,0), (0,1).
    • There was a Youtube on the topic (Blue2Brown1).
    • Scalar - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
    • ℝⁿ - A vector space of n-dimensions over the field ℝ. ℝ is the scalar type. ℝ², ℝ³, etc...

Common Groups

  • Dihedral Groups: Dₙ - D₃ is the dihedral group of the triangle.
  • Symmetry Groups: Sₙ - S₃ the symmetric group of 3 elements. S₃ is isomorphic to D₃. (S₃ ≅ D₃)
  • Orthogonal group: O(n) - Reflections & rotations. The group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations (∘). It is equivalent to the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.
  • Special Orthogonal: SO(n) - Rotation group. SO(3) is the group of all rotations about the origin of three-dimensional Euclidean space ℝ³ under the operation of composition (∘). eg Roll, Pitch, Yaw. SO(n) ≤ O(n).
  • General Linear: GLₙ(ℝ) - The group of all n×n matrices with non-zero determinants under matrix multiplication. nonabelian group because matrix operations are non-commutative. The set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. The group GLₙ(ℝ) over the field of real numbers is a real Lie group of dimension n².
  • Special Linear: SL(n) - Is the subgroup of GL(n, 𝔽) consisting of matrices with a determinant of 1.
  • ΓLₙ(ℝ) - General semilinear group. Contains GL.

Common Lie Algebras

  • 𝔰𝔬(3) - The Lie algebra of SO(n) and would be the rate of change for Roll, Pitch and Yaw. 𝔰𝔬(n) is equal to 𝔬(n) - Princeton Mathematics (p234)
  • 𝔤𝔩ₙ(ℂ) is the Lie algebra for GLₙ(ℂ), the space of all n×n complex matrices.
  • 𝔰𝔩ₙ(ℂ) - The Lie algebra of the special linear group SLₙ(n). "Is the subspace of all matrices with trace zero" - Princeton Mathematics (p234)

Cayley table

A table showing all the results of all possible operations on a finite group.

Eignvectors

  • Eigenvector - A vector in a vector space doesn't get rotated by a linear transformation. It stays on it's 'span'. It's the axis of rotation.
  • Eigenvalue - The value an Eigenvector is scaled by. 𝛢𝓋⃗=λ𝓋⃗. 𝛢 is transformation matrix. 𝓋⃗ is an Eigenvector. λ is the eigenvalue.


Lie Stuff

  • Like calculus for groups?

Lie Group - A finite, continuous group. - Symmetry and the monster, pg62.

Analysis

Calculus

Gradients

Misc

  • Axiom of choice - Given a non-empty collection of non-empty sets you can form a new set by picking one element from each set. - Elementary Topology and it's applications pg20.

Math Computer Programming

>>> from primesieve.numpy import *

# Generate a numpy array with the primes below 100
>>> generate_primes_array(100)

Videos

MOOCs

YouTube


Unsorted

HOW TO LEARN ADVANCED MATHEMATICS WITHOUT HEADING TO UNIVERSITY - PART 1 PART 3

4chan - Math Textbook Recommendations

Harvard Course of Abstract Algebra (apparently goes well with the Artin book)

List of books


Order of abstract algebra


This guide on algebra

This guide to imaginary numbers

Math intuition

These guides


These notes are recommended

HOW TO LEARN MATH: FOR STUDENTS

Books

To Read

  • Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table - Looks like a mid level introduction to stuff.

Reading

  • What is Mathematics? - Seems like a good overall math introduction. Has non-stupid exercises.
  • The Foundations of Mathematics by Ian Stuart. - It's mostly just about logic, set theory, proofs, etc...
  • Symmetry and the Monster - Finished. Didn't have too much learning content but wasn't a long read. Forgotten a lot of it already though...

Misc

  • Steven Strogatz
    • The Joy of X
    • Sync
    • Nonlinear Dynamics
    • The Calculus of Friendship


  • The Mathematical Mechanic - Mark Levi - Using Physics to solve math problems. Mentioned on a 3Blue2Borown YouTube video.
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