Mathematical Constructs

Mathematics is a formal precise language we have invented; It is essentially a short form and disciplined way of writing logical arguments.  (18, July 2015)

Every mathematical object parametrized by n real numbers may be treated as a point of the n-dimensional space of all such objects.   (Bernhard Riemann in 1854)

———————————————————-

 

Set

A group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. (C=A.B)

 

  • ∀u,v,w∈V: u+(v+w)=(u+v)+w
  • ∀u∈V: u+0=u
  • ∀u∈V ∃−u∈V: u+(−u)=0

 

An abelian group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. (commutativity, AoB=BoA)

 

  • ∀u,v∈V: u+v=v+u

 

A ring is an abelian group with a second binary operation that is associative (Ao(BoC)=(AoB)oC), is distributive over the abelian group operation (Ao(B.C)=(AoB).(AoC)) and has an identity element

A field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

————————————————————————————

 

A vector space is an abelian group of vectors, which may be multiplied (“scaled”) by numbers, called scalars in this context.

 

  • ∀a∈? ∀u,v∈V: a⋅(u+v)=(a⋅u)+(a⋅v)
  • ∀a,b∈? ∀u∈V: (a+b)⋅u=(a⋅u)+(b⋅u)
  • ∀a,b∈? ∀u∈V: a⋅(b⋅u)=(ab)⋅u
  • ∀u∈V: 1?⋅u=u

 

A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a subspace of V.

Topology:

http://en.wikipedia.org/wiki/Simplex

http://en.wikipedia.org/wiki/Simplicial_complex

http://en.wikipedia.org/wiki/Betti_number

Loading