Jim Wendelberger Science and Mathematics Blog

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.

Permutation Groups by Martin W. Liebeck.

Phase Transitions.

 p

Orbifolds.

Ordinals.

The Peano Axioms.

Optimization and Lagrange Multipliers by Keith Ball.

Optimization.

The Gradient and Contours.

Constrained Optimization and Lagrange Multipliers.

The General Method of Lagrange Multipliers.

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.

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.

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

Modular Forms.

A Lattice in the Complex Numbers.

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.

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.