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Neutron multiplicity analysis (NMA) is a non-destructive method for estimating fissile mass from the rate of detecting one, two or more emitted fission neutrons within a time gate, known as the neutron moments. NMA relies on the point model which expresses the neutron moments as analytic expressions of a constant and uniform probability for each neutron to induce fission. To use a constant probability of fission the point model approximates the geometry of assemblies of fissile materials as a single point in time and space, neglecting all geometry. This approximation significantly underestimates the fissile mass and enrichment for large assemblies. This work presents the Varying Probability of Fission (VPF) model, which extends the point kinetic model by allowing the probability of fission to vary with position and the number of fissions in each fission chain, termed the neutron collision number. While the VPF model requires a several values of the fission probability, additional constraints can be imposed to reduce the number of solved variables. These constraints can be modelled interpolation functions with parameters determined from Monte Carlo simulation. Additionally, the Associated Particle Imaging (API) technique provides additional information on interrogating neutron direction and transmission images. After applying a back projection step API systems provide imaged data of emitted neutron singles and doubles for induced fission at varying positions in addition to the transmission image. This work presents how the imaged data of neutron transmission, neutron singles and doubles can be incorporated into a VPF analysis and provide images of the spatially dependent induced neutron fission probability and the rate of induced neutron chains. The VPF model could enable a new, real-time quantitative assessment capability for nuclear material assemblies that are imaged by API systems.

Integer compositions and related enumeration problems have been extensively studied. The cyclic analogues of such questions, however, have significantly fewer results. In this thesis, we follow the cyclic construction of Flajolet and Soria to obtain generating functions for cyclic compositions and n-color cyclic compositions with various restrictions. With these generating functions we present some statistics and asymptotic formulas for the number of compositions and parts in such compositions. Combinatorial explanations are also provided for many of the enumerative observations presented.

The 0-1 Knapsack Problem is an NP-hard optimization problem that has been studied extensively since the 1950s, due to its real world significance. The basic problem is that a knapsack with a weight capacity c is to be filled with a subset of n items. Each item i, has a weight value wi and a profit value pi. The goal is to maximize total profit value without the having the total weight exceed the capacity.In this thesis, the 0-1 Knapsack Problem is introduced and some of the research and applications of the problem are given. Pisinger's branch-and-bound algorithm that will converge to an optimal solution is presented. One of the earliest applications of the knapsack problem, the knapsack cryptosystems, is then discussed. The earliest knapsack cryptosystem, the Merkle-Hellman Cryptosystem, is described along with how Adi Shamir broke this cryptosystem. Generating functions are then used to provide a number of solutions to a knapsack problem. Using the generating function of the knapsack problem, the paper concludes with an application on the Electoral College.

In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one type. Sheffer extensively developed properties of the B-Type 0 polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary B-Type k by constructing the generalized generating function A(t)exp[xH1(t) + · · · + xk+1Hk(t)] = ∑∞n=0 Pn(x)tn, with Hi(t) = hi,iti + hi,i+1t i+1 + · · · , h1,1 ≠ 0. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher (k ≥ 1). We present a preliminary analysis of a special case of the B-Type 1 (k = 1) class, which is an extension of the B-Type 0 class, in order to determine which sets, if any, are also orthogonal sets. Lastly, we consider an extension of this research and comment on future considerations. In this work the utilization of computer algebra packages is indispensable, as computational difficulties arise in the B-Type 1 class that are unlike those in the B-Type 0 class.

We consider sequences of nonnegative integers $S=(s_1,s_2,ldots, s_n)$ defined by systems of constraints represented as weighted directed graphs in which edge $(s_a,s_b)$ of weight $w$ indicates constraint $s_a~geq~s_b~+~w$. We propose a set of seven rules and a decomposition technique for obtaining multivariate and single-variable generating functions for families of such graphs. Our method compares to existing techniques by offering an elegant and intuitive approach to obtaining generating functions and recurrences, albeit only for a subset of the partition and composition enumeration problems addressed by other techniques. The decomposition technique we propose remains relevant, nevertheless, to a wide range of applications, including several well-known ones. Moreover, our objective is to obtain recurrences for generating functions so as to assist the formulation and proof of their closed-form solutions. For integer sequences defined by directed graphs with $w in 0,1]$, we prove that our technique holds sufficient. We describe the formulation of finite-variable generating function recurrences from multi-variable ones and provide a set of rules to determine the variables chosen. The construction tree is introduced as the tree representation of the construction of a generating function from the decomposition of a weighted directed graph. Given such a construction tree, automation of the process of building the multi-variable and finite-variable recurrences is possible, and we implement it as a computer program. Finally, we apply our techniques and tools to a wide range of famous problems, including 2-rowed plane partitions, up-down compositions and hexagonal plane partitions, as well as some new problems, and obtain recurrences to each. We find that our methods are not only effective but also easy and simple to use.

We prove that, for any fixed d, there is a polynomial time algorithm for computing the generating function of any projection of the set of integer points in a d-dimensional polytope. This implies that many interesting sets of integer points can be encoded as short rational generating functions, such as the Frobenius semigroup of all nonnegative integer combinations of given positive integers, affine semigroups, neighbors and the neighborhood complex (also known as the Scarf complex or complex of maximal lattice-free bodies), Hilbert bases, and sets from algebraic integer programming. We also show how to use the generating functions to solve computational problems (such as finding the cardinality of the set or finding its maximum element) in polynomial time. We may also use this theorem to compute, as a short rational function, the Hilbert series of rings generated by monomials. We examine the connection between generating functions and the neighborhood complex, and we consider possibilities for improving the algorithm for the main theorem. Finally, we examine the relationship between rational generating functions and the complexity of Presburger arithmetic.