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.
Archive for September, 2010
III.91 Transforms by T. W. Körner: Pages 303 – 304.
Thursday, September 30th, 2010III.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 (R2n, w0).
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 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, 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.”