Maths - Notation

Multidimensional Algebra

notation	meaning	basis	Alternative
	direct sum or Kronecker sum depending on context
	direct product or Kronecker product depending on context
/\	exterior algebra
/\k	exterior algebra kth blade
	scalar or real depending on context (see below)	e0
²	2 dimensional vector	e1,e2
³	3 dimensional vector	e1,e2,e3
n	n dimensional vector	e1…en
^²²	bivector based on 2D algebra	e12
^²³	bivector based on 3D algebra	e12,e31,e23
C	complex numbers
H	quaternions		³
O	octonions		7
aA	a is an element of the set A

Scalar

I have used the term 'scalar' interchangeably with the term 'real', that is, a continuous value that can be represented by a single number.

Strictly speaking the term 'scalar' should be reserved for a quantity that is used to scale a vector, that is change its magnitude without changing its direction, or in other words a scalar is the ratio of parallel vectors.

For instance I should not really call energy a scalar because there are no vectors involved.

I apologise for my lack of mathematical rigor here, its just that the word scalar seems to better express that it is not a vector and its less likely to cause confusion with the real part of a complex number. Also this (mis?)usage is quite common in the computer world.

Matrix

notation	meaning
[M]	matrix
[M]t	transpose of matrix (swap rows & columns)
[M]-1	inverse of matrix
[I]	identity matrix (ones on leading diagonal, otherwise zeros)

The individual elements of the matrix are numbered as follows,

m00	m01	m02	…	m0n
m10	m11	m12	…	m1n
m20	m21	m22	…	m2n
mp0	mp1	mp2	…	mpn

The first subscript represents the row, the second subscript represents the column

Tensor Notation

notation	example	meaning	pages in site
subscript	e1	coordinate basis	tensors
superscript	x1	coordinate value
Einstein Summation Convention	eixi=e1x1+e2x2+e3x3=∑eixi	When the same index appears twice in an expression, once raised and once lowered, a sum is implied.
partial derivatives	∂a=∂/∂a

Set Theory

notation	example	meaning
A	A = {a | aA}	set
a	aA	element of A
{a, b, c}		set containing a, b and c
	a H = {ah | hH}	left coset of a
(a,b)		cycle notation
<a,b>		all the elements of group generated by a and b
	<r,f | r³=1, f²=1, frf=r-1>	generators with constrains
K	K = {k | kK}	complement of set indicated by bar over top
| A |		grade (size) = number of elements in A
| G : H |		number of left cosets of H in G
×		Direct Product
		Semidirect Product (left)
	{n, h}φ {n',h'}	Semidirect Product (right)
		Bicrossed Product also known as Knit or Zappa-Szep product
	h = h'H such that a h' = h a	Normal subgroup a H = H a for all a in G
Ø		empty set
¬		not
∧		and (= intersection)
∨		or (U = union)
		is in (is an element of the set)
	AB	A is contained in B (or A is a subset of B)
->		if
<->		if and only if
		for all
		there exists
		is isomorphic to
(S)		The set of subsets of S (power set)
orb(s)		Orbit - set of elements that can be reached
stab(s)		Stabilizer - set of elements that don't move s

Sets are discussed on this page and groups here.

Definition

Endo is a Greek word meaning inside.

Logic / Proofs

notation	example	meaning	pages in site
/\		conjunction, and , intersection (in set), greatest lower bound (in order)
\/		disjunction, or , union (in set), least upper bound (in order)
T		Top (true)
_|_		Bottom (false)
=>	A => B	Implication A => B means A implies B
	Tφ	φ is a theorem of T that is: φ is deducible from T In contrast to 'implication' above this is a statement about statements usually written only when we mean φ follows from T, wheras A => B is a statement we may consider as a conjecture . See [Lawvere,Rosebrugh page 198] Not to be confused with right adjoint in category theory.	logic
		for all (quantifier)
		there exists (quantifier)

Category Theory

notation	example	meaning	pages in site
AB		Map / Morphism / Function between mathematical structures that relates or preserves the structures in some way. May be injection, surjection or bijection (injection can be specifically indicated see below)	category theory maps/morphisms
a b		A function in terms of elements of the structures aA, bB
		A function that is valid if all the functions drawn as solid lines are valid
		injective morphism (embedding)
		canonical map - map of G onto factor group G/H where H is a normal subgroup of G
		Left adjoint	adjunctions
		Right adjoint

Topology Botanical Notation

Topology Notation

Denotional Semantics

notation	example	meaning	pages in site
[[D]]	[[2*5]] = 10	A 'denotion' is written in double square brackets. It represents the meaning of D to a program evaluation of D	domain theory denotional semantics
		Bottom (undefined)
		Less defined than

Greek Alphabet

	upper	lower		upper	lower		upper	lower
Alpha	A	α	Iota	I	ι	Rho	P	ρ
Beta	B	β	Kappa	K	κ	Sigma	Σ	σ
Gamma	Γ	γ	Lambda	Λ	λ	Tau	Τ	τ
Delta	Δ	δ	Mu	M	μ	Upsilon	Υ	υ
Epsilon	E	ε	Nu	N	ν	Phi	Φ	φ
Zeta	Z	ζ	Xi	Ξ	ξ	Chi	X	χ
Eta	H	η	Omicron	O	ο	Psi	Ψ	ψ
Theta	Θ	θ	Pi	Π	π	Omega	Ω	ω

Various Operator Symbols

×		÷		±		•
°		ð		Ð		ƒ		µ
Ø		ø		Þ		þ		†
		…		∂		◊		≤
≥		∈		≠		≈		∫
∞		∀		∃		√

Powers of 10

m	10-3	K	103
µ	10-6	M	106
n	10-9	G	109
p	10-12	T	1012
f	10-15
a	10-18