Maths: Difference between revisions

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* [https://en.wikipedia.org/wiki/Golden_ratio_base phinary] in base-φ
* [https://en.wikipedia.org/wiki/Golden_ratio_base phinary] in base-φ
* [https://en.wikipedia.org/wiki/Non-integer_representation Non-integer representation]
* [https://en.wikipedia.org/wiki/Non-integer_representation Non-integer representation]
* <math>\Sigma_A</math> - The permutation group of set A.


=Axioms=
=Axioms=

Revision as of 21:40, 4 April 2017

Misc

desmos geometry

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

NumPy cheatsheet

Project Euler - Math challenges.

A variation on an Ulam Spiral: a Sacks Spiral

More on Stacks Spiral

Parabola point

Notation and Symbols

Alphanumeric

  • Superscript: 𝓍⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ ᴬᴮᴰᴱᴳᴴᴵᴶᴷᴸᴹᴺᴼᴾᴿᵀᵁⱽᵂ
  • Subscript: 𝓍₀₁₂₃₄₆₇₈₉ ₊₋₌₍₎ ₐₑₒₓₔₕₖₗₘₙₚₛₜ ᵢᵣᵤᵥₓ ᵦᵧᵨᵩᵪ
  • Blackboard Bold: 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ 𝕒𝕓𝕔𝕕𝕖𝕗𝕘𝕙𝕚𝕛𝕜𝕝𝕞𝕟𝕠𝕡𝕢𝕣𝕤𝕥𝕦𝕧𝕨𝕩𝕪𝕫 𝟘𝟙𝟚𝟛𝟜𝟝𝟞𝟟𝟠𝟡 ℾℽℿℼ⅀ ⅅⅆⅇⅈⅉ
  • Script: 𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵 𝒶 𝒷 𝒸 𝒹 ℯ 𝒻 ℊ 𝒽 𝒾 𝒿 𝓀 𝓁 𝓂 𝓃 𝓅 𝓆 𝓇 𝓈 𝓉 𝓊 𝓋 𝓌 𝓍 𝓎 𝓏
  • Script (Bold): 𝓐𝓑𝓒𝓓𝓔𝓕𝓖𝓗𝓘𝓙𝓚𝓛𝓜𝓝𝓞𝓟𝓠𝓡𝓢𝓣𝓤𝓥𝓦𝓧𝓨𝓩𝓪𝓫𝓬𝓭𝓮𝓯𝓰𝓱𝓲𝓳𝓴𝓵𝓶𝓷𝓸𝓹𝓺𝓻𝓼𝓽𝓾𝓿𝔀𝔁𝔂𝔃
  • Fraktur: 𝔄𝔅ℭ𝔇𝔈𝔉𝔊ℌℑ𝔍𝔎𝔏𝔐𝔑𝔒𝔓𝔔ℜ𝔖𝔗𝔘𝔙𝔚𝔛𝔜ℨ𝔞𝔟𝔠𝔡𝔢𝔣𝔤𝔥𝔦𝔧𝔨𝔩𝔪𝔫𝔬𝔭𝔮𝔯𝔰𝔱𝔲𝔳𝔴𝔵𝔶𝔷
  • Fraktur (Bold): 𝕬𝕭𝕮𝕯𝕰𝕱𝕲𝕳𝕴𝕵𝕶𝕷𝕸𝕹𝕺𝕻𝕼𝕽𝕾𝕿𝖀𝖁𝖂𝖃𝖄𝖅𝖆𝖇𝖈𝖉𝖊𝖋𝖌𝖍𝖎𝖏𝖐𝖑𝖒𝖓𝖔𝖕𝖖𝖗𝖘𝖙𝖚𝖛𝖜𝖝𝖞𝖟
  • Serif (Bold): 𝐀𝐁𝐂𝐃𝐄𝐅𝐆𝐇𝐈𝐉𝐊𝐋𝐌𝐍𝐎𝐏𝐐𝐑𝐒𝐓𝐔𝐕𝐖𝐗𝐘𝐙𝐚𝐛𝐜𝐝𝐞𝐟𝐠𝐡𝐢𝐣𝐤𝐥𝐦𝐧𝐨𝐩𝐪𝐫𝐬𝐭𝐮𝐯𝐰𝐱𝐲𝐳
  • Sans-serif: 𝖠𝖡𝖢𝖣𝖤𝖥𝖦𝖧𝖨𝖩𝖪𝖫𝖬𝖭𝖮𝖯𝖰𝖱𝖲𝖳𝖴𝖵𝖶𝖷𝖸𝖹𝖺𝖻𝖼𝖽𝖾𝖿𝗀𝗁𝗂𝗃𝗄𝗅𝗆𝗇𝗈𝗉𝗊𝗋𝗌𝗍𝗎𝗏𝗐𝗑𝗒𝗓
  • Sans-Serif (Bold): 𝗔𝗕𝗖𝗗𝗘𝗙𝗚𝗛𝗜𝗝𝗞𝗟𝗠𝗡𝗢𝗣𝗤𝗥𝗦𝗧𝗨𝗩𝗪𝗫𝗬𝗭𝗮𝗯𝗰𝗱𝗲𝗳𝗴𝗵𝗶𝗷𝗸𝗹𝗺𝗻𝗼𝗽𝗾𝗿𝘀𝘁𝘂𝘃𝘄𝘅𝘆𝘇
  • Mathematical Italic: 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧𝛩
  • Mathematical Italic (bold): 𝑨𝑩𝑪𝑫𝑬𝑭𝑮𝑯𝑰𝑱𝑲𝑳𝑴𝑵𝑶𝑷𝑸𝑹𝑺𝑻𝑼𝑽𝑾𝑿𝒀𝒁𝒂𝒃𝒄𝒅𝒆𝒇𝒈𝒉𝒊𝒋𝒌𝒍𝒎𝒏𝒐𝒑𝒒𝒓𝒔𝒕𝒖𝒗𝒘𝒙𝒚𝒛
  • ıȷ

Links

Latex

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\,}

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: ∑ π φ 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.
  • {♠, ♦, ♥, ♣}
  • ↪ - Embedding... 𝒻: 𝐗↪𝐘. 𝒻 is the identity map, taking 𝓍∈𝐗 to 𝓍∈𝐘 and 𝐗⊂𝐘. Maybe 'lift'.
  • ↣ - 𝒻: 𝐗↣𝐘 is an injective function from 𝐗 to 𝐘. Injection.
  • ↠ - 𝒻: 𝐗↠𝐘, ↠ is a surjection (possibly only an 'onto' surjection. See also this. and pdf
  • ≡ - Congruence. 23 ≡ 45 (mod 11)
  • |𝐴|×|𝐵|

Elementary Algebra

Goodish Overview

Number Systems & Groups

  • 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.
  • ℙⁿ - n-dimensional projective space. ℝⁿ/scaling. If you can rescale one vector to another, they are the same.
  • 𝔸ⁿ - Affine Space. A n-dimensional real vector space without origin. ℝⁿ/translations. 2 Objects points and vectors. ~3:00min in
  • ℤ/pℤ - The set of integers modulo some prime p. Or just ℤ/p for short.
  • p-adic - Changes the measurement metric to one modulus a prime.
  • ℓ-adic - 'el-adic'. - New Theories Reveal the Nature of Numbers - Étale cohomology
  • Wikipedia: Non-standard positional numeral systems
  • 𝒢ₙ, 𝒢(p,q) - The geometric algebra generated by the vector space of signature (p,q) is 𝒢(p,q). 𝒢ₙ refers to all of them. 𝒢(2,0) / 𝒢(3,0) is a 2D/3D Euclidean algebra. "Geometric Algebra for Physicists".
  • phinary in base-φ
  • Non-integer representation

Axioms

List of axioms

Peano Arithmetic

The axioms that define the natural numbers. - Good Math pg5.

  • Initial Value Rule - There is one object called 0 and 0 is a natural number.
  • Successor rule: For every natural number n there is exactly one other natural number called its sucessor, s(n).
  • Uniqueness Rule: No two natural numbers have the same sucessor.
  • Equality Rules: Numbers can be compared for equality.
    • Equality is reflexive: - Every number is equal to itself
    • Equality is symmetric: a=b then b=a
    • Equality is transitive: if a=b and b=c then a=c
  • Induction rule: For P, P is true for all natural numbers if.
    • 1. P is true about 0. P(0)=true.
    • 2. If you assume P is true for a natural number n(P(n) is true). Then you can prove that P is true for the sucessor s(n) of n, P(s(n) == true.

Addition

  • Commutative: n+m = m+n
  • Identity: n+0 = 0+n=n
  • Recursion: m+s(n) = s(m + n)

Naive Set Theory

∀𝓍(𝓍 ∈ 𝓎 → φ(𝓍))

Alternative Axiomatic Set Theories The Axioms of Set Theory

ZFC Axioms

The new axiom of set theory and Bell inequality.

There are a few differnt symbolisms of it. Jech and Kunen seem to be popular.

Wikipedia: Zermelo–Fraenkel set theory [http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html Wolfram MathWorld: Zermelo-Fraenkel Axioms

Axiom of empty set

Seems to be missing from many versions of ZFC since it can apparently be derived from the axiom of infinity. Wikipedia: https://en.wikipedia.org/wiki/Axiom_of_empty_set.

  • ∃𝓍∀𝓎¬(𝓎 ∈ 𝓍) - There is a set that no set is a member of it.
  • ∃∅∀𝓍(𝓍∉∅)
  • ∃∅ : ∀𝓍¬∈ ∅

1. Axiom of extensionality

∀𝓍∀𝓎[∀𝓏(𝓏 ∈ 𝓍 ⇔ 𝓏 ∈ 𝓎) ⇒ 𝓍 = 𝓎 - Sets are equal if they share the same elements.

Algebraic Structures

Properties

  • associative - 𝒶(𝒷𝒸) = (𝒶𝒷)𝒸. ∀𝓍,𝓎,𝓏. 𝓍∘(𝓎∘𝓏)=(𝓍∘𝓎)∘𝓏
  • commutative - 𝒶𝒷 = 𝒶𝒷. ∀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. The identity element of multiplication is 1 as 1×𝓍=𝓍=𝓍×1. The identity of addition is 0 as 0+𝓍=𝓍=𝓍+0.
  • 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. The multiplicative inverse: 𝒶𝒶⁻¹=1=𝒶⁻¹𝒶


  • domain - The set of elements on which a function has a valid definition is it's domain. For 𝒻: X→Y, X is the domain.
  • codomain / target - The possible set that a function outputs. For 𝒻: X→Y, X is the target. There may be things in the target that aren't actually reachable by the function.
  • image / range - The set of elements that a function maps to and no more. This is like the target but only the possible elements are in the set. For example the cos function could be defined as having a target of the real numbers, but the range could only be between -1 and 1. It is not always possible to define the range. Concepts of Modern Mathematics (pg. ~67). Some ambiguity as sometimes 'range' refers to the codomain.


  • 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 referred to as a linear map.
  • injection - Each target element must be reachable from only one element in the domain. It does not need to be surjective, meaning that it's possible to have elements in the target that aren't reachable at all. But you can not have an element in the domain that is reachable from more than one target element. cos is not an injection because an infinite number of inputs get mapped to values between -1 and 1.
  • surjection (onto) - If every element of a target set T is reachable by a function, that function is a surjection onto T. This means the target is also the range. Does not need to be an injection. - Concepts of Modern Mathematics (pg. 70).
  • bijection - A bijection is invertible. It is an injection and a surjection. A function which relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S. A mapping that is both one-to-one (an injection) and onto (a surjection).
  • 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]

In ℝ[x] we can take congruents to the modulus 𝓍²+1. Two polynomials are congruent if their difference is divisible by 𝓍²+1. This makes the polynomial ring behave like complex numbers. When using a 'prime' polynomial as modulus, the polynomial ring is a field.

𝓍³+𝓍²-2𝓍+3 ≡ -3𝓍+2 (mod 𝓍²+1)

Mentioned in a concrete guide to modern mathematics, page 90.

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.

  • 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).

ℚ, ℝ, ℂ are fields.

ℤₚ is a field if p is prime.

ℤ is not a field.

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

  • Aₙ - Alternating Group - The group of even permutations of a finite set.
  • Dₙ - Dihedral Groups - A dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. D₃ is the dihedral group of the triangle. D₁≅Z₂, D₂≅K₄.
  • Sₙ - Symmetry Groups - S₃ the symmetric group of 3 elements. S₃ is isomorphic to D₃. (S₃ ≅ D₃)
  • O(n) - Orthogonal group - 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.
  • SO(n) - Special Orthogonal - Rotation group. SO(2) is the circle 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. Non-commutative. SO(n) ≤ O(n).
  • POₙ(ℝ) - Projective orthogonal group. PSO(V) for projective special orthogonal group.
  • GLₙ(𝔽) - General Linear - 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 as the set of all n×n real matrices, Mₙ(ℝ), forms a real vector space of dimension n².
  • Special Linear: SL(n) - Is the subgroup of GL(n, 𝔽) consisting of matrices with a determinant of 1. These form a group because the product of two matrices with determinant 1 again has determinant 1.
  • ΓLₙ(ℝ) - General semilinear group. Contains GL.
  • Sp(n) - The compact symplectic group Sp(n) is often written as USp(2n), indicating the fact that it is isomorphic to the group of unitary symplectic matrices.
  • Sp(2n, 𝔽) - The symplectic group of degree 2n over a field 𝔽 is the group of 2n × 2n symplectic matrices with entries in 𝔽, and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, 𝔽).
  • Uₙ(ℝ) - The unitary group - A group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SUₙ(ℝ).
  • 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 \Sigma_A} - The permutation group of set A.


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

Geometry

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

Reading

  • Fearless Symmetry - Good book.
  • What is Mathematics? - Seems like a good overall math introduction. Seems to have non-stupid exercises.
  • The Princeton Companion to Mathematics - Covers basically everything.
  • Good Math - Seems like a basic approachable introduction.
  • The Foundations of Mathematics by Ian Stuart. - It's mostly just about logic, set theory, proofs, etc... Not badly written but the concepts aren't that interesting. Probably give up?
  • 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...

To Read

  • Mathematician's Delight - Several recommendations. Apparently a good introduction to maths.
  • A Mathematician's Apology
  • How to Solve It - Bunch of recommendations. (also some books on TODO).
  • Pi the next generation - Seems cool. Can calculate pi to any arbitrary digit.
  • The Elements of Mathematics - Seems to be a Princeton book.
  • Handbook of Practical Logic and Automated Reasoning - Recommended for programmers.
  • Concrete mathematics - Knuth
  • The Princeton Companion to Applied Mathematics


Topology 🍩

  • Topology - Jänich - Seems approachable while in depth.
  • Elementary Applied Topology - Robert Ghrist - Seems not too bad. Lots of pictures and starts with manifolds.
  • Experiments in Topology - Seems not too bad again.

Groups

  • Groups and Symmetry: A Guide to Discovering Mathematics - The tile maps puzzle.
  • Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table - Looks like a mid level introduction to stuff.
  • Group Theory for Physicists

Misc

  • Steven Strogatz
    • The Joy of X
    • Sync
    • Nonlinear Dynamics
    • The Calculus of Friendship
  • W. W.-Sawyer - Old but apparently wrote decent introduction books.
  • The Mathematical Mechanic - Mark Levi - Using Physics to solve math problems. Mentioned on a 3Blue2Borown YouTube video.
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