Archive for September, 2010

III.91 Transforms by T. W. Körner: Pages 303 – 304.

Thursday, September 30th, 2010

Transforms by T. W. Körner.  If a is a sequence a0, a1, a2, …, we can define an “infinite polynomial” (Ga)(t) to be S¥r=0 artr.  We need to worry about in what sense this sum exists.  Formally, (Ga)(t)(Gb)(t) = (G(a*b))(t), where the infinite sequence c = a * b is given by ck = a0bk + a1bk-1 + … + akb0 .  We call this the convolution of a and b.

III.90 Topological Spaces by Ben Green: Pages 301 – 303.

Wednesday, September 29th, 2010

Topological Spaces by Ben Green.

The discrete topology, Euclidean spaces, subspace topology, the Zariski topology, connectedness and compactness.

Relationship to continuity.

III.89 Tensor Products: Page 301.

Tuesday, September 28th, 2010

Tensor Products.

Vector spaces over a field and a bilinear map.

Can be generalized to any algebraic structure for which bilinearity makes sense such as a module or a C*-algebra.

III.88 Symplectic Manifolds by Gabriel P. Paternain: Pages 299 – 300.

Sunday, September 26th, 2010

Symplectic Manifolds by Gabriel P. Paternain.

Gromov’s Nonsqueezing Theorem.

Symplectic Manifolds.

III.88 Symplectic Manifolds by Gabriel P. Paternain: Page 298.

Friday, September 24th, 2010

Symplectic Manifolds by Gabriel P. Paternain.

Symplectic Diffeomorphisms of (R2n, w0).

Hamiltons Equations.

III.88 Symplectic Manifolds by Gabriel P. Paternain: Page 297.

Thursday, September 23rd, 2010

Symplectic Manifolds by Gabriel P. Paternain.

Sympletic Linear Algebra.

The standard sympletic form has the properties of: bilinearity, antisymmetry and nondegenerativity.

III.87 Spherical Harmonics: Pages 296 -297.

Wednesday, September 22nd, 2010

A spherical harmonic of order n and dimension d is the restriction to the sphere Sd-1 of a harmonic polynomial in d variables that is homogeneous of degree n.

 Spherical harmonics link Chebyshev and Legendre polynomials.

III.87 Spherical Harmonics: Page 295.

Tuesday, September 21st, 2010

Spherical Harmonics.

III.86 The Spectrum by G. R. Allen: Pages 294 – 295.

Monday, September 20th, 2010

The Spectrum by G. R. Allen.

Generalization of eigenvalues.

III.85 Special Functions by T. W. Körner: Pages 291 – 293.

Thursday, September 16th, 2010

Special Functions by T. W. Körner.

“In practice, a ‘special function’ is any function that, like the logarithm and gamma function, has been extensively studied and has turned out to be useful.”