Bessel Functions - Function Trigonometric Fourier Series - Orbit Stabiliser Theorem - Mathematics Assignment Help

Assignment Task :

Information and instructions
1. You should attempt all 10 questions.
2. Your assignment should be submitted online via moodle by 3rd August 2020, and should be either completed by hand or typed up using LATEX.
3. This is an individual assignment rather than group work. Collaborating on the assignment will be regarded as cheating.
4. This assignment is worth 100% of the August coursework mark.
5. To obtain full marks, in addition to being technically correct your work must be clear and your method explained fully.

1. Recall that Bessel functions can be expressed as
 

2. Parameterise the line formed by the intersection of a two spheres, one of radius r1 centred on (1, 0, 0) and one of radius r2 centred on (-1, 0, 0).

6. Split each edge of the square in half and label each half-edge clockwise using elements of the set X = f1, 2, 3, 4, 5, 6, 7, 8g. The elements of the symmetry group
D8 permute the elements of X.
(a) List the 8 permutations of X, writing each in cycle notation.
(b) Find the order of each element.
(c) Show that the action of D8 on X is proper.

8. (a) Prove that Fp[[t]], the ring of formal power series over the finite field of order prime p, is an integral domain.
(b) Prove that Fp[[t]] is not a field.

9. Use the Orbit-Stabiliser Theorem in two different ways to find the order of the group of symmetries of the icosahedron (pictured).

You may assume that
(a) the action of G is transitive on the set of 12 vertices and that the outer edges of the 5 adjacent faces of a vertex form a regular pentagon.
(b) the action of G is transitive on the set of 20 faces, each of which forms an equilateral triangle.

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