In this thesis we study distribution properties of values of certain polynomial maps at integral points. The first set of values that we investigate is the values of a linear map at integral points on a quadratic surface. we establish conditions sufficient to ensure that this set is dense and' then under stronger conditions we show that this set is equidistributed. The methods employed in this situation depend on the uniform distribution of unipotent flows on homogeneous spaces. In particular we use Ratner's theorems. The second set of values that we explore is the values of a quadratic form at integral points. We show that under certain dimension and rationality conditions the n-point correlations of this set behave in a way which is consistent with the behaviour of the correlations of uniformly distributed set of random numbers. The methods used in this setting are based on Fourier analysis and certain asymptotic estimates for theta series.

This thesis is concerned with the characteristics and behaviours of random polynomi- als of a high degree. Random polynomials are polynomials with random coefficients and take the form '£']=0 ajxj, where the coefficients aj, (j = 0, ... , n) are random variables of a probability distribution, such as the normal or uniform distribution. The polynomials featured in this thesis are of the algebraic, hyperbolic and trigonometric type, all of which have coefficients that are independent random variables of the nor- mal distribution. In Chapter 1, we discuss the characteristics featured throughout this thesis, namely, the expected number of level crossings, the expected number of maxima( minima), the expected number of maxima below a fixed level u, the expected number of points of inflection as well as the covariance of the number of zeros. In this chapter we also present the formulae used in this thesis to prove the results obtained. In Chapter 2 we discuss results previously obtained for polynomials with similar characteristics to those featured in this thesis.

This thesis studies the mathematical behavior of random polynomials in terms of the expected number of real zeros and the exceedance measure. To this end, the important formulae for studying this behavior are reviewed, generalized and applied to random polynomials of three different types: the polynomials with symmetric coefficients, the amplified polynomials and the de-amplified polynomials. By defining the coefficients a/ s from the classic form of polynomial P(x) = L 7=oa jxi as independently normally distributed random variables with two-fold symmetry, aj = a-n-j-1 a polynomial with symmetric coefficients is studied. Then by introducing binomial factors, (n,j) 1/2, into coefficients, the types of amplified and de-amplified polynomials are defined. Besides the expected number of real zeros, the exceedance measure is considered for better understanding of the behavior of amplified and de-amplified polynomials. Later in this thesis a valued progress is made towards a conjecture that constants been missed from the sum of binomial series have insignificant impact on the expected number of real zeros. The early studies in this thesis implies that the binomial factors facilitate the evaluation of the behavior, therefore a more difficult class of random polynomial without binomial factors is studied as final part of work of the thesis and supported by numerical analysis

In this thesis we study the invariant rings for the Sylow p-subgroups of the finite classical groups. We have successfully constructed presentations for the invariant rings for the Sylow p-subgroups of the unitary groups GU(3, IF,,) and GU(4, IF,, ), the symplectic group Sp(4,IF,) and the orthogonal group Q+(4,IF,) with q odd. In all cases, we obtained a minimal gene~ating set which is also a SAGEl basis. Moreover: we computed the relations among the generators and showed that the invariant ring for these groups are a complete intersection. This shows that, even though the invariant rings of the Sylow p-subgroups of the general linear group are polynomial, t he same is not true for Sylow p-subgroups of general classical groups. We also constructed the generators for the invariant fields for the Sylow p-subgroups of GU(n,IF,,), Sp(2n,IF,), Q+(2n,IF,), Q-(2n + 2, IF,) and Q(2n + 1, IF,), for every n and q. This is an important step in order to obtain the generators and relations for the invariant rings of all these groups.

This thesis consists of two parts. In the first part an elliptic generalisation of the Bernoulli polynomials is introduced and investigated. We first consider the Faulhaber polynomials which are simply related to the even Bernoulli polynomials and generalise them in relatwn with the classical Lamé equation using the integrals of the Korteweg-de-Vries equation. An elliptic version of the odd Bernoulli polynomials is defined in relation to the quantum Euler top. These polynomials are applied to compute the Lamé spectral polynomials and the densities of states of the Lamé operators. In the second part we consider a special class of periodic continued fractions that we call α-fractions.

The chromatic polynomials of certain families of graphs can be calculated by a transfer matrix method. The transfer matrix commutes with an action of the symmetric group on the colours. Using representation theory, it is shown that the matrix is equivalent to a block-diagonal matrix. The multiplicities and the sizes of the blocks are obtained. Using a repeated inclusion-exclusion argument the entries of the blocks can be calculated. In particular, from one of the inclusion-exclusion arguments it follows that the transfer matrix can be written as a linear combination of operators which, in certain cases, form an algebra. The eigenvalues of the blocks can be inferred from this structure. The form of the chromatic polynomials permits the use of a theorem by Beraha, Kahane and Weiss to determine the limiting behaviour of the roots. The theorem says that, apart from some isolated points, the roots approach certain curves in the complex plane. Some improvements have been made in the methods of calculating these curves. Many examples are discussed in detail. In particular the chromatic polynomials of the family of the so-called generalized dodecahedra and four similar families of cubic graphs are obtained, and the limiting behaviour of their roots is discussed.

In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and the equation being unsolvable by radicals may appear contradictory when read out of context. Secondly, the reason why some mathematical software programs may fail to render a conclusive test of the formula, and how that can be corrected is explained. As an application, this formula is used to determine another formula that expresses the gravitational constant in terms of other known physical constants. It is also explained why up to now it has been impossible to determine this expression using the current underlying theoretical basis. / M. Sc. (Applied Mathematics)

In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and the equation being unsolvable by radicals may appear contradictory when read out of context. Secondly, the reason why some mathematical software programs may fail to render a conclusive test of the formula, and how that can be corrected is explained. As an application, this formula is used to determine another formula that expresses the gravitational constant in terms of other known physical constants. It is also explained why up to now it has been impossible to determine this expression using the current underlying theoretical basis. / M. Sc. (Applied Mathematics)