The cycle polynomial of a finite permutation group G is the generating function for the number of elements of G with a given number of cycles:

In this thesis, basic knot theory is introduced, along with concepts from topology, algebra and algebraic topology, as they relate to knot theory. In the first chapter, basic definitions concerning knots are presented. In the second chapter, the fundamental group is applied as a method of distinguishing knots. In particular the torus knots are classified using the fundamental group, and a gener...

0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

This article outlines techniques for computing the Galois group of a polynomial over the rationals, an important operation in computational algebraic number theory. In particular, the linear resolvent polynomial method of [6] will be described.

We compute a polynomial with Galois group SL 2 (11) over Q.Furthermorewe prove that SL 2 (11) is the Galois group of a regular extension of Q(t).

We consider integrals of type ∫ On u1 11 . . . u an 1nu b1 21 . . . u bn 2n du, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials. Introduction The computation of polynomial integral...

Oblivious polynomial evaluation is a protocol involving two parties, a sender whose input is a polynomial P , and a receiver whose input is a value x. At the end of the protocol, the receiver learns P (x) and nothing more about P , while the sender remains oblivious of both x and P (x). It is used as a primitive in many applications including protocols for private comparison of data, for mutual...

We present a functional encryption scheme based on standard assumptions where ciphertexts are associated with a tuple of values (x1, . . . , xn) ∈ Zp , secret keys are associated with a degree-two polynomial, and the decryption of a ciphertext ct(x1,...,xn)∈Znp with a secret key skP∈Zp[X1,...,Xn],deg(P )≤2 recovers P (x1, . . . , xn), where the ciphertext contains only O(n) group elements. Our ...