Gosh math sure is elegant.

(Source: photosystemii)

## Peano Axioms

“In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers.

The constant 0 is assumed to be a natural number:

1. 0 is a natural number.

The next four axioms describe the equality relation.

2. For every natural numberx,x=x. That is, equality isreflexive.
3. For all natural numbersxandy, ifx=y, theny=x. That is, equality issymmetric.
4. For all natural numbersx,yandz, ifx=yandy=z, thenx=z. That is, equality istransitive.
5. For allaandb, ifais a natural number anda=b, thenbis also a natural number. That is, the natural numbers areclosedunder equality.

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued “successor” function S.

6. For every natural numbern,S(n) is a natural number.

Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define aunary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.

7. For every natural numbern,S(n) = 0 is false. That is, there is no natural number whose successor is 0.

8. For all natural numbersmandn, ifS(m) =S(n), thenm=n. That is,Sis aninjection.

Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S(0), S(S(0)), and furthermore that {0, S(0), S(S(0)), …} ⊆ N. This shows that the set of natural numbers is infinite. However, to show that N = {0, S(0), S(S(0)), …}, it must be shown that N ⊆ {0, S(0), S(S(0)), …}; i.e., it must be shown that every natural number is included in {0, S(0), S(S(0)), …}. To do this however requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.

9. If K is a set such that:

• 0 is in K, and
• for every natural number n, if n is in K, then S(n) is in K,

then K contains every natural number.

The induction axiom is sometimes stated in the following form:

9. If φ is a unary predicate such that:

• φ(0) is true, and
• for every natural number n, if φ(n) is true, then φ(S(n)) is true,

then φ(n) is true for every natural number n.

In Peano’s original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section Models below”

(Source: Wikipedia)

## Vector Spaces

A Vector space is a set of Vectors, V, a set of scalars (known as a Field) as well as the rules for Vector addition, and the multiplication of a Vector by a scalar.

(V,F,+,*)

Examples of sets of vectors:

• set of Polynomials
•  ℝn (n euclidean  space) the plane is ℝ2
• the set of all nxm, Matrices (where n and m are constant).

Examples of sets of Fields:

•  (the set of real numbers)
•  (the set of complex numbers)
Example of the rule for vector addition in 2:
v, w ℝ2 ie
v = [x1, x2], w = [y1,y2]
v+w = [x1 + y1, x2 + y2]
Example of the rule for vector multiplication in 2:
v∈ 2 , z
v = [x1,x2]
z*v = [z*x1,z*x2]

References:

van den Heuvel, M. P., Hulshoff Pol, H. E., 2010. Exploring the brain network: A review on resting-state fMRI functional connectivity. European Neuropsychopharmacology 20, 519-553.www.tumblr.com/tumblelog/jee-neuro/new/photo

Bullmore, E., Sporns, O., 2009. Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience 10, 186-198

(via barrelshifter)

## 10/30/12

I just learned about n-cubes/hypercubes and they are beautiful.

I finally understand why we were taught Gray code… I love when concepts connect!

…Also, today I ran into the girl who had to wait behind me in line at the grocery store last night. At first I didn’t know why a stranger was trying to talk to me but apparently she recognized me (!!!) and said “Hey, weren’t you at the grocery store yesterday? With the yogurt?” Weird, what are the chances.

When two numbers a and b are natural,

and the greatest common factor (GCF) of (a,b) = 1, then a and b are said to be Reletivly prime.

The Handshaking Theorem:

The total degree of a graph is twice the number of edges.