General

Binomial theorem:

From Discrete mathematics and its applications:

            n  
(x + y)ⁿ =  Σ  ⁿCⱼ(xⁿ⁻ⁱ)(yⁱ)
           i=0


         = ⁿC₀(xⁿ) + ⁿC₁(xⁿ⁻¹)(y) + ⁿC₂(xⁿ⁻²)(y²) + .... +  ⁿCₙ(yⁿ)

From this, it follows that, the coefficient of xⁱyʲ is ⁿCⱼ

For example, coefficient of x¹²y¹³ in (x+y)²⁵ is:

  ⁿCᵢ


= ²⁵C₁₃ 


        25!            
=  --------------  = 5200300
    (25-13)! 13!

Conjecture

A statement believed to be true but which hasn't been proven yet.

Eg: Goldbach's conjecture

Homomorphism

f(a+b) = f(a) + f(b)

where + denotes the homomorphism.

Structure preserving mapping.

Eg: z ↦ z mod 2 is structure preserving (but also loses some info.)

Isomorphism

• Structure preserving mapping with an inverse that is also structure preserving.
• ie, every isomorphism is also a homomorphism.
• Unlike homomorphism, info will not be lost.
  • Always preserves all info.

https://math.stackexchange.com/questions/731724/what-is-the-difference-between-homomorphism-and-isomorphism

Morphism

In category theory, morphisms are also known as arrows. ³

Endomorphism

A morphism from a mathematical object to itself.

An endomorphism that is also an isomorphism is an automorphism.

Linear map

A mapping from a vector space V to another vector space W (ie, V → W), that preserves the operations of vector addition and scalar multiplication.²

If the mapping is bijective, then the linear map is a linear isomorphism.

If V = W, the linear map is a linear endomorphism.

Commensurability

Two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise they are incommensurable.

Examples:

• √3/2 : incommensurable
• 2√3/√3: commensurable

(Note: commensurability in group theory is something different)

'Weirdness' of limits

      1
    ----- = 1 + x + x² + x³ + ...
     1-x

=>    1   = (1-x)(1 + x + x² + x³ + ...)
=>    1   = (1 + x + x² + x³ + ...) - (x + x² + x³ + ...)
=>    1   = 1

is not what it looks like.

Try the original equation with x = 2.

      1
    ----- = 1 + x + x² + x³ + ...
     1-x


      1
=>  ----- = 1 + 2 + 2² + 2³ + ...
     1-2


=>   -1   = 1 + 2 + 2² + 2³ + ...

which is wrong.

Try taking sum of first n+1 elements.

         n
Sₙ = 1 + Σ  
        i=1

2 + 4 + 8 + 16 + 32 + 2 6 14 30 64

Reference: Calculus for scientists and engineers: An analytical approach - K. D. Joshi

Infinitesimal number

A number other than zero that is closer to zero than all other real numbers.

Relation between limits and derivatives

Limit

lim f(x) = v
x→k

When value of x tends to k, value of f(x) tends to v.

Continuous function

A function f(x) is continuous at a point c if whenever Δx is infinitesimaly small, f(c+Δx) - f(c) is infinitesimaly small as well.

 f(c+Δx) - f(c)
---------------  = f'(c)
     f(c)

where f'(c) is the first derivative of f(x) at c.

Hyperreal number system

A number system with all the real numbers and infinitesimal numbers.

Some properties of limits

If

lim aₙ = A   and    lim bₙ = B
n→∞                 n→∞             

then,

lim (aₙ + bₙ) = A + B           (sum rule)
n→∞

lim (aₙ - bₙ) = A - B           (difference rule)
n→∞

lim (k.aₙ)    = k.A             (constant multiple rule)
n→∞

lim (aₙ * bₙ) = A * B           (product rule)
n→∞

lim (aₙ / bₙ) = A / B, if B≠0   (division rule)
n→∞

Reference: Thomas' Calculus (12e) - George B. Thomas Jr., Maruice D. Weir, Joel R. Hass

Meaning of limit tending to infinity

lim aₙ = ∞
n→∞

does not mean that aₙ approaches to ∞.

(Converging to ∞ means difference between the terms of aₙ and ∞ becomes smaller and smaller. But that cannot happen as ∞ is, well, infinity.)

It's just a notation (ie, for limits in the case of ∞).

aₙ eventually gets and stays larger than any fixed number as n gets large.

Reference: Thomas' Calculus (12e) - George B. Thomas Jr., Maruice D. Weir, Joel R. Hass

Perfect power

https://en.wikipedia.org/wiki/Perfect_power

Basically means that a number n can be expressed in aᵇ form where, a>1, b>1, a∈ℕ, b∈ℕ

If n = aᵏ, then n can be called a perfect k-th power.

• Perfect square is when k=2 (eg: 36 = 6²)
• Perfect cube is when k=3 (eg: 125 = 5³)

Conjectures

Goldbach's conjecture

Every even natural number greater than 2 can be expressed as the sum of two prime numbers.

Some webages: l1, l5

Twin prime conjecture

aka Polignac’s conjecture.

There are infinitely many twin primes.

(Twin primes are prime numbers that differ by 2. Like 3 & 5.)

Diophantine equations

A polynomial equation with

• integer coefficients
• a finite number of variables

where

• only solutions of interest are integers

Eg:

• x² - y² = z³
• x⁴ + y⁴ + z⁴ = w⁴
• ax + by = c (a linear Diophantine equation)

See Britannica page

Fundamental theorems

• Fundamental theorem of arithmetic: Every integer greater than 1 can be represented as a unique prime factorization.
• Fundamental theorem of calculus: Integration and differentiation are inverses of each other

Fundamental theorem of algebra

• aka d'Alembert's theorem
  • (Not to be confused with d'Alembert's principle in physics.)

From Britannica:

every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers.

Remember: ℝ ⊂ ℂ

And: ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ ²⁰

Fermat's last theorem

Named after 17th century French mathematician Pierre de Fermat.

For a,b,c ∈ ℤ⁺ and n ∈ ℤ,
(ie, a, b, c are positive integers.)

there is no n > 2 such that

aⁿ + bⁿ = cⁿ

is satisfied.

ie, the largest integer for which this equation is true is 2.

Fun fact: with n=2, we have the equation corresponding to the Pythagorus theorem for right triangles:

a² + b² = c²

Statement of this theorem was discovered 30 years after Fermat's death and hence the name by which it became known.

No proof of this theorem was discovered in Fermat's writings, and numerous unsuccessful attempts were made to produce a proof. So much that it was considered more as a conjecture for many years till a proof was finally discovered in 1994 by the English mathematician Andrew Wiles.

Fractals

Sierpiński curve

  +        +
 / \      / \
+   +----+   +
 \          /
  +        +
  |        |
  +        +
 /          \
+   +----+   +
 \ /      \ /
  +        +

Sierpiński carpet

https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet

• 3D counterpart: Monger sponge

+---+---+---+---+---+---+---+---+---+
|111|111|111|111|111|111|111|111|111|
+---+---+---+---+---+---+---+---+---+
|111|   |111|111|   |111|111|   |111|
+---+---+---+---+---+---+---+---+---+
|111|111|111|111|111|111|111|111|111|
+---+---+---+---+---+---+---+---+---+
|111|111|111|           |111|111|111|
+---+---+---+           +---+---+---+
|111|   |111|           |111|   |111|
+---+---+---+           +---+---+---+
|111|111|111|           |111|111|111|
+---+---+---+---+---+---+---+---+---+
|111|111|111|111|111|111|111|111|111|
+---+---+---+---+---+---+---+---+---+
|111|   |111|111|   |111|111|   |111|
+---+---+---+---+---+---+---+---+---+
|111|111|111|111|111|111|111|111|111|
+---+---+---+---+---+---+---+---+---+

Multiset

• Like set, but duplicate elements possible.
• Each element in a multiset has a multiplicity associated with it, which is simply the number of times that element shows up in the multiset.
• Order of elements doesn't matter
  • Eg: {1,1,2,2,2} ≡ {1,1,2,2,2} ≡ {1,1,2,2,2}

Eg:

• {1,1,2,2,2} is a multiset where
  • Multiplicity(1) = 2
  • Multiplicity(2) = 3

Archimedean property/norm

Any two non-zero values are comparable.

ie, no two values are infinitesimely apart with respect to each other.

There are no infinitely large or infinitely small numbers.

Totient function

aka Euler's totient function.

φ(n) is the number of natural numbers less than n which are co-prime to n. These natural numbers are then called the totatives of n.

Eg:

• φ(9) = |{1,2,4,5,7,8}| = 6
• φ(8) = |{1,3,5,7}| = 4

sage: f=sloane.A000010
sage: f
Euler's totient function

sage: f(9)
6
sage: a=9; a.coprime_integers(a)
[1, 2, 4, 5, 7, 8]

sage: f(8)
4
sage: a=8; a.coprime_integers(a)
[1, 3, 5, 7]

Vandermonde matrix ʷ

• Named after ther French mathematician Alexandre-Théophile Vandermonde.

A matrix in which each row is a geometric progression.

⎡ 1  a1  a1²  ...  a1ⁿ ⎤
⎢ 1  a2  a2²  ...  a2ⁿ ⎥
⎢     ....             ⎥
⎢     ....             ⎥
⎣ 1  am  am²  ...  amⁿ ⎦

ie, aᵢⱼ = aᵢʲ⁻¹ (where indexing starts from 1)* Geometry

• Parallelepiped (/ˌparəlɛləˈpʌɪpɛd/ or /ˌparəlɛˈlɛpɪpɛd/): A solid object whose each face is a parallelogram. Like a generalized form of cube.

—

Determinant non-zero (ie, non-singular) as long as aᵢ-s are distinct.

But ppk was talking about the transpose of the matrix that I got from wikipedia.

Congruency vs similarity

Similarity means the proportions are conserved. Congruency means the measures (angles, sides, etc) should be equal.

All congruent triangles are similar, but not the other way around.

Disc

A disc is a region of a plane bounded by a circle.

Closed disc

The disc includes the bounding circle.

Open disc

Bounding circle is not part of the disc.

Polygon

• Polygon actually consists only of the 'boundary' and not the region inside it.

(Reminds one of a polynomial, doesn't it..)

Euler's identity ʷ

e^{iπ} + 1 = 0

ie, e^iπ is (almost?) -1.

>>> pow(math.e, math.pi * 1j)
(-1+1.2246467991473532e-16j)

where e is Euler's number.

    ∞         1     2     3    
e = Σ  = 1 + --- + --- + --- + ....
   i=0        1!    2!    3!   

Group theory

Quotient set

Quotient of a set A by a relation R is the set of all equivalence classes induced on A by R.

Could think of all the 'equivalent' elements of A being considered single elements, and the set of all these 'merged' elements is the quotient set. ¹⁴

Eg:

A = ℕ
R x = x mod 5

Then, quotient set of A with respect to R is:

{[0], [1], [2], [3], [4]}

Order of a group

• Number of elements present in the set associated with the group (ie, cardinality of the set) for finite groups.
• In the case of infinite groups, order is infinite.

Odd order theorem

every finite group of odd order is solvable. ¹⁶

• aka Feit-Thompson theorem
• A proof made with Coq exists.
• DBT: What's there to solve in a group?
  • https://mathworld.wolfram.com/SolvableGroup.html

SL(2,ℤ) in group theory

Simply put, set of all 2x2 integer matrices whose determinant is 1.

Set of a matrices M

     ⎡a  b⎤
M =  ⎢    ⎥
     ⎣c  d⎦

such that a, b, c, d ∈ ℤ

and

|M| = 1

From here:

⎧                              ⎫
⎮⎡a  b⎤                        ⎮
⎨⎢    ⎥ | a,b,c,d ∈ ℤ, ad-bc=1 ⎬
⎮⎣c  d⎦                        ⎮
⎩                              ⎭

Not yet sure why this set is of special interest though…

Order theory

Partially ordered set

• aka poset
• Consists of a set together with a binary relation.
• The relation: partial ordering
  • The name signifies that not every pair of elements are comparable (unlike total orderings).

Totally ordered set

• A set equipped with a total ordering.
• Could think of as a 'stronger' partial order, where every pair of elements are comparable.
• aka:
  • Linearly ordered set (loset)
  • Simply ordered set

Total ordering (a relation) is usually written as ≤. It is:

• Reflexive: if a ∈ A then a ≤ a
• Anti-symmetric: if a b ∈ A, a ≤ b and b ≤ a, then a = b
• Transitive: if a b c ∈ A, a ≤ b and b ≤ c, then a ≤ c
• Strongly connected: if a b ∈ A, then either a ≤ b or b ≤ a

Total ordering is also known as:

• linear order
• full order

Lattice

A lattice is defined on a poset.

• Every pair of elements have:
  • a unique supremum (aka join)
  • a unique infimum (aka meet)
• Join (⊔)
  • Specific to a given pair of elements of the poset.
  • aka least upper bound, aka supremum
• Meet (⊓)
  • Specific to a given pair of elements of the poset.
  • aka greatest lower bound, aka infimum

Example lattices:

• ℕ ordered by divisibility
  • Join: LCM
  • Meet: GCD

Bounded lattice

Lattice is bounded if there exists a top element ⊤ ∈ A and a bottom element ⊥ ∈ A such that,

∀x ∈ A,

  x ⨆ ⊤ = ⊤
  x ⨅ ⊥ = ⊥

Topology

Informally called:

• 'Rubber sheet geometry'
• 'Elastic geometry'

From How to cut a cake by Ian Stewart:

Geometry of continuous transformations.

provides a language of continuity that is general enough to include a vast array of phenomena while being precise enough to be developed in new ways.

• Möbius strip

• Klein bottle

• Continuity

• Relative position matters, not magnitudes

  • Hence the nick name 'rubber sheet geometry'.

Kinds of topology

• Point-set topology
• Algebraic topology

Metric space

A space where there is a definition of the notion of distance.

ie, distance between any pair of points in the space is defined.

For this, the following conditions must be true:

• d(x, y) ≥ 0
• d(x, y) = 0 means x=y
• d(x, y) = d(y, x)
• d(x,y) + d(y,z) ≥ d(x,z)
  • Because the straight path is always the shortest.

where X is the metric space and x,y,z ∈ X.

Cobordism

An n-dimensional manifold going through a manifold of n-1 dimension.

Homotopy

Map between maps??

2 maps f0,f1: X → Y are homotopic if there exists a homotopy connecting them.

ie, f0 value at any t ∈ [0,1] can be mapped to f1 value at that t???

f0 and f1 homotopic means: f0 ≃ f1

Homotopy equivalence

A map f:X→Y is called a homotopy equivalence if there is another map g:Y→X such that

fg ≃ 𝟙 and gf ≃ 𝟙

Then,

• f and g have the same homotopy type.
• Spaces X and Y are homotopy equivalent
  • Notation: X ≃ Y

Fun fact: For two topological spaces X and Y, X ≃ Y is an equivalance relation.

• Any space is homotopy equivalent to itself. (reflexivity)
• If X ≃ Y, then Y ≃ X (symmetric)
• If X ≃ Y and Y ≃ Z, then X ≃ Z (transitivity)

Path

A path in a space X is a continuous map f:[0,1] → X.

I guess we could think of it as the path followed by a point with I=[0,1] as indicative of time.
f gives the position of the point at each time instant.

Another guess: 2 paths are said to be homotopic if one path can brought to overlap with the other by means of a map f, where the end-points of the paths are the same.

Winding number ʷ

Winding number of a closed curve with respect to a point in a plane is the number of counter-clockwise loops the curve takes around that point.

Clockwise turns => negative.

Misc

New terms

• Homology

• Homology type

• Homotopy

• Closed sets

• Open sets

• Dense sets

• Quotient space

• Homeomorphism: 'a correspondence between two figures or surfaces or other geometrical objects, defined by a one-to-one mapping that is continuous in both directions.' - from Britannica

Notations:

• a ~ b means a is related b (ie, (a,b) ∈ R where R is a relation).

References:

• A first course in topology - John McCleary

Linear algebra

Basis of a vector space

Set of vectors B where every element of the vector space can be written as a finite linear combination of (elements of) B. ʷ

• Basis vectors: elements of a basis
  • Basis vectors are linearly independent.
• Dimension of vector space: number of elements in basis

Eg:

ℝ² is a vector space under component-wise addition and scalar multiplication.

ie,

(a,b) + (c,d) = (a+b,c+d)

λ.(a,b) = (λa,λb)

Following is a basis of ℝ²:

e0 = [1, 0]
e1 = [0, 0]

∵ any vector (a,b) ∈ ℝ² can be written as ae₀ + be₁.

Linearly independent vectors

A set of vectors v₁, v₂, .. , vₙ are said to be linearly dependent ʷ if there is some (non-trivial) linear combination of them that equals the zero vector.

v₁, v₂, .. , vₙ are linearly dependent if there exists scalars a₁, a₂, .. , aₙ (where at least one aᵢ is non-zero) such that:

a₁v₁, a₂v₂, .. , aₙvₙ = 0

If no such set of scalars exist, then the set of vectors are linearly independent.

import sympy
import numpy as np
a=np.array([[1,-3,2],
            [1,2,4]])
aa = sympy.Matrix(a)
aa.rref()
#
# (Matrix([
# [1, 0, 16/5],
# [0, 1,  2/5]]), (0, 1))
#
# ie, 2 vectors linearly independent but the other is dependent

Characteristic polynomial ʷ

Characteristic polynomial of a square matrix.

Tensors vs vectors

Tensors are basically a generalization of scalars, vectors and matrices.

Order/rank of a tensor is 'defined by the number of directions required to describe it'. ʳ

More: https://physics.stackexchange.com/questions/257838/what-is-the-difference-between-a-tensor-vector-and-a-matrix

Factoids

• Zero vector: all components zero

Get the first column of a 2x2 matrix:

Just multiply it with [1 0]ᵀ.

Eg:

⎡ 2  3 ⎤ ⎡ 1 ⎤   ⎡ 2 ⎤
⎢      ⎥ ⎢   ⎥ = ⎢   ⎥
⎣ 4  5 ⎦ ⎣ 0 ⎦   ⎣ 4 ⎦

Tips

• All right isosceles triangles are similar.
• Group ℤₙ is associated with the set {0,1,2, … , n-1} with addition mod n as the operation.
• Power tower: https://mathworld.wolfram.com/PowerTower.html
  • 'Height' of the tower