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!
A statement believed to be true but which hasn't been proven yet.
Eg: Goldbach's conjecture
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.)
In category theory, morphisms are also known as arrows. ³
A morphism from a mathematical object to itself.
An endomorphism that is also an isomorphism is an automorphism.
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.
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:
(Note: commensurability in group theory is something different)
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
A number other than zero that is closer to zero than all other real numbers.
lim f(x) = v
x→k
When value of x tends to k, value of f(x) tends to v.
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.
A number system with all the real numbers and infinitesimal numbers.
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
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
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.
Every even natural number greater than 2 can be expressed as the sum of two prime numbers.
| 4 | 2 + 2 |
| 6 | 3 + 3 |
| 8 | 5 + 3 |
| 10 | 7 + 3 |
| 100 | 53 + 47 |
aka Polignac’s conjecture.
There are infinitely many twin primes.
(Twin primes are prime numbers that differ by 2. Like 3 & 5.)
A polynomial equation with
where
Eg:
See Britannica page
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
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.
+ +
/ \ / \
+ +----+ +
\ /
+ +
| |
+ +
/ \
+ +----+ +
\ / \ /
+ +
https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet
+---+---+---+---+---+---+---+---+---+
|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|
+---+---+---+---+---+---+---+---+---+
Eg:
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.
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:
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]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
/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.
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.
A disc is a region of a plane bounded by a circle.
(Reminds one of a polynomial, doesn't it..)
e^{iπ} + 1 = 0
ie, eiπ 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!
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]}
every finite group of odd order is solvable. ¹⁶
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…
Total ordering (a relation) is usually written as ≤. It is:
a ∈ A then a ≤ aa b ∈ A, a ≤ b and b ≤ a, then a = ba b c ∈ A, a ≤ b and b ≤ c, then a ≤ ca b ∈ A, then either a ≤ b or b ≤ aTotal ordering is also known as:
A lattice is defined on a poset.
⊔)
⊓)
Example lattices:
Lattice is bounded if there exists a top element ⊤ ∈ A and a bottom element ⊥ ∈ A such that,
∀x ∈ A,
x ⨆ ⊤ = ⊤
x ⨅ ⊥ = ⊥
Informally called:
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
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:
where X is the metric space and x,y,z ∈ X.
An n-dimensional manifold going through a manifold of n-1 dimension.
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
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.X and Y are homotopy equivalent
X ≃ YFun fact: For two topological spaces X and Y, X ≃ Y is an equivalance relation.
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 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.
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:
Set of vectors B where every element of the vector space can be written as a finite linear combination of (elements of) B. ʷ
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.
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 dependentCharacteristic polynomial of a square matrix.
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
| Tensor | ||
| order | A representation | Dimensionality |
| 0 | Scalar | 0 |
| 1 | Vector | 1 |
| 2 | Matrix | 2 |
Just multiply it with [1 0]ᵀ.
Eg:
⎡ 2 3 ⎤ ⎡ 1 ⎤ ⎡ 2 ⎤
⎢ ⎥ ⎢ ⎥ = ⎢ ⎥
⎣ 4 5 ⎦ ⎣ 0 ⎦ ⎣ 4 ⎦