Archive for July, 2010

III.70 Pi: Page 262.

Saturday, July 31st, 2010

Defining properties of p.

All defining properties of p are simple but they do not lead to solutions of polynomial equations. Thus it is not surprising that p is both irrational and transcendental.

III.68 Permutation Groups by Martin W. Liebeck, III.69 Phase Transitions, III.70 Pi: Pages 259 – 261.

Friday, July 30th, 2010

Permutation Groups by Martin W. Liebeck.

Phase Transitions.

 p

III.65 Orbifolds, III.66 Ordinals, III.67 The Peano Axioms: Page 258.

Thursday, July 29th, 2010

Orbifolds.

Ordinals.

The Peano Axioms.

III.64 Optimization and Lagrange Multipliers by Keith Ball: Pages 255 – 257.

Wednesday, July 28th, 2010

Optimization and Lagrange Multipliers by Keith Ball.

Optimization.

The Gradient and Contours.

Constrained Optimization and Lagrange Multipliers.

The General Method of Lagrange Multipliers.

III.63 Number Fields: Page 254.

Tuesday, July 27th, 2010

Number Fields.

A number field K is a finite-degree field extension of the field of rational numbers.

The simplest are the quadratic fields.

A number is an algebraic integer of the number field if it is a root of a monic polynomial with coefficients in the usual Integers, Z.  One may embed fields into Ideals.  The Ideals maintain the unique factorization into integers.

Dirichlet’s unit theorem.

III.62 Normed Spaces and Banach Spaces: Page 253.

Monday, July 26th, 2010

A Banach Space is a complete normed space.

Named after Stefan Banach 1892 – 1945.

In contrast a Hilbert Space is a complete inner product space.

A metric may be used to define the a norm.

III.59 Modular Forms by Kevin Buzzard, General Lattices, Relations between Lattices, Modular Forms as Functions on Lattices, and Why Modular Forms, III.60 Moduli Spaces, III.61 The Monster Group, III.62 Normed Spaces and Banach Spaces: Pages 251-252.

Sunday, July 25th, 2010

Modular Forms by Kevin Buzzard

General Lattices

Relations between Lattices

Modular Forms as Functions on Lattices

Why Modular Forms

Moduli Spaces

The Monster Group

The number 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 appears!

Normed Spaces and Banach Spaces

III.59 Modular Forms by Kevin Buzzard, A Lattice in the Complex Numbers: Page 250.

Friday, July 23rd, 2010

Modular Forms.

A Lattice in the Complex Numbers.

III.57 Models of Set Theory, III.58 Modular Arithmetic by Ben Green: Pages 249 – 250.

Thursday, July 22nd, 2010

Models of set theory.

The model in which the usual axioms, ZF or ZFC, of set theory hold.

The model of ZF is a set.

Modular arithmetic.

The proof of Fermat’s little theorem.

Euler’s theorem.

III.56 Metric Spaces: Page 248.

Wednesday, July 21st, 2010

Metric Spaces

A set of points X and a metric d, a function of two points in X, that satisfies the properties:

d is non-negative and zero if the two points are identical,

d is symmetric and

d satisfies the triangle inequality,

is called a metric space.