Jim Wendelberger Science and Mathematics Blog

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 a_rt^r.  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 c_k = a₀b_k + a₁b_k-1 + … + a_kb₀ .  We call this the convolution of a and b.

Topological Spaces by Ben Green.

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

Relationship to continuity.

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.

Symplectic Manifolds by Gabriel P. Paternain.

Gromov’s Nonsqueezing Theorem.

Symplectic Manifolds.

Symplectic Manifolds by Gabriel P. Paternain.

Symplectic Diffeomorphisms of (R²ⁿ, w₀).

Hamiltons Equations.

Symplectic Manifolds by Gabriel P. Paternain.

Sympletic Linear Algebra.

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

A 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.

Spherical Harmonics.

The Spectrum by G. R. Allen.

Generalization of eigenvalues.

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.”