Transforms by T. W. Körner. If **a** is a sequence a_{0}, a_{1}, a_{2},
, we can define an “infinite polynomial” (G**a)**(t) to be S^{¥}_{r=0} a_{r}t^{r}. We need to worry about in what sense this sum exists. Formally, (G**a**)(t)(G**b)**(t) = (G(**a*****b**))(t), where the infinite **sequence c** **= a** *** b** is given by c_{k} = a_{0}b_{k} + a_{1}b_{k-1}^{ }+
+ a_{k}b_{0} . We call this the convolution of **a** and **b**.

## Archive for September, 2010

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

Thursday, September 30th, 2010### III.90 Topological Spaces by Ben Green: Pages 301 – 303.

Wednesday, September 29th, 2010Topological 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, 2010Tensor 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, 2010Symplectic Manifolds by Gabriel P. Paternain.

Gromov’s Nonsqueezing Theorem.

Symplectic Manifolds.

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

Friday, September 24th, 2010Symplectic Manifolds by Gabriel P. Paternain.

Symplectic Diffeomorphisms of (R^{2n}, w_{0}).

Hamiltons Equations.

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

Thursday, September 23rd, 2010Symplectic 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, 2010A spherical harmonic of order n and dimension d is the restriction to the sphere S^{d-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, 2010Spherical Harmonics.

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

Monday, September 20th, 2010The Spectrum by G. R. Allen.

Generalization of eigenvalues.

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

Thursday, September 16th, 2010Special 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.”