We provide a combinatorial derivation of the exponential growth constant for counting sequences of lattice path models restricted to the quarter plane. The values arise as bounds from analysis of related half planes models. We give explicit formulas, and the bounds are provably tight. The strategy is easily generalized to cones in higher dimensions, and has implications for random generation.

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.

Topics in torsion theory STELIOS CHARALAMBIDES The purpose of this thesis is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being r-torsion and r-torsionfree modules as well as r-dense and r-pure submodules. In the first chapter we also introduce a new concept, that of a r-compact module, which is basic enough to deserve a place among the preliminaries. The results that we obtain fall into three areas which are to a certain degree interrelated. The first area is on r-Max modules, which we introduce as a torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsion-theoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left r-Max rings and apply the torsion-theoretic version of Shock's theorem to obtain a characterization of r-short modules motivated by a recent article in which short modules were introduced. The second area deals with various flavours of r-injectivity, some known and some new. We introduce r-Al-injective and s-r-A'l-injective modules and examine their relationship with the known concepts of r-injective and r-quasi-injective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-r-AI-injective modules, and give a generalization of Azumaya's Lemma. We also prove that every M-generated module has a r-Al-injective hull which is unique up to isomorphism and show how this is linked to the r-quasi-injective hull. We then examine E-r-injectivity, generalizing well-known results by Faith, Albu and Nastasescu and Cailleau which provide necessary and sufficient conditions for the E-s-r-injective property, the E-s-r-A'l-injective property and for a direct sum of E-s-r-A4-injective modules to be E-s-r-A'f-injective. In the third area we introduce some new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. The approach is twofold. The first is via r-CS modules which serve as a generalization of CS modules

In this article we consider injective modules relative to a torsion theory τ. We introduce τ-M-injective and s-τ-M-injective modules, relatively τ-injective modules, the τ-M-injective hull and Σ-τ-M-injective and Σ-s-τ-M-injective modules. We then examine the relationship between these new and known concepts. Some of the new results obtained include a Generalized Fuchs Criterion characterizing s-τ-M-injective modules, a Generalized Azumaya's Lemma characterizing τ-I M iinjective modules, the proof of the existence and uniqueness up to isomorphism of the τ-M-injective hull and generalizations of results by Albu and Nȃstȃsescu, Faith, and Cailleau on necessary and sufficient conditions for a module to be Σ-s-τ-M-injective, Σ-τ-injective and for a direct sum of Σ-s-τ-M-injective modules to be Σ-s-τ-M-injective. To Robert Wisbauer on the occasion of his 65th birthday.

We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in dimensions 4 and 5 and we also give some examples from higher dimensions. This construction generalizes easily to each complex simple Lie algebra.

We define an integrable hamiltonian system of Toda type associated with the real Lie algebra so(p, q). As usual there exists a periodic and a non-periodic version. We construct, using the root space, two Lax pair representations and the associated Poisson tensors. We prove Liouville integrability and examine the multi-hamiltonian structure. The system is a projection of a canonical An type Toda lattice via a Flaschka type transformation. It is also obtained via a complex change of variables from the classical Toda lattice.

In 1974, Johnson showed how to multiply and divide sparse polynomials using a binary heap. This paper introduces a new algorithm that uses a heap to divide with the same complexity as multiplication. It is a fraction-free method that also reduces the number of integer operations for divisions of polynomials with integer coefficients over the rationals. Heap-based algorithms use very little memory and do not generate garbage. They can run in the cpu cache and achieve high performance. We compare our C implementation of sparse polynomial multiplication and division with integer coefficients to the routines of existing computer algebra systems.

In 1974, Johnson showed how to multiply and divide sparse polynomials using a binary heap. This paper introduces a new algorithm that uses a heap to divide with the same complexity as multiplication. It is a fraction-free method that also reduces the number of integer operations for divisions of polynomials with integer coefficients over the rationals. Heap-based algorithms use very little memory and do not generate garbage. They can run in the cpu cache and achieve high performance. We compare our C implementation of sparse polynomial multiplication and division with integer coefficients to the routines of the Magma, Maple, Pari, Singular and Trip computer algebra systems.

We present a new algorithm for pseudo-division of sparse multivariate polynomials with integer coefficients. It uses a heap of pointers to simultaneously merge the dividend and partial products, sorting the terms efficiently and delaying all coefficient arithmetic to produce good complexity. The algorithm uses very little memory and we expect it to run in the processor cache. We give benchmarks comparing our implementation to existing computer algebra systems.

We demonstrate new routines for sparse multivariate polynomial multiplication and division over the integers that we have integrated into Maple 14 through the expand and divide commands. These routines are currently the fastest available, and the multiplication routine is parallelized with superlinear speedup. The performance of Maple is significantly improved. We describe our polynomial data structure and compare it with Maple's. Then we present benchmarks comparing Maple 14 with Maple 13, Magma, Mathematica, Singular, Pari, and Trip.

The algorithms for linear algebra in the Magma and Axiom computer algebra systems work over an arbitrary ring. For example, the implementation of Gaussian elimination for reducing a matrix to (reduced) row Echelon form works over any field that the user constructs. In contrast, Maple's facilities for linear algebra in its LinearAlgebra package only work for specific rings. If the input matrix contains general expressions, the algorithms may work incorrectly.

We present a high performance algorithm for multiplying sparse distributed polynomials using a multicore processor. Each core uses a heap of pointers to multiply parts of the polynomials using its local cache. Intermediate results are written to buffers in shared cache and the cores take turns combining them to form the result. A cooperative approach is used to balance the load and improve scalability, and the extra cache from each core produces a superlinear speedup in practice. We present benchmarks comparing our parallel routine to a sequential version and to the routines of other computer algebra systems.

A common way of implementing multivariate polynomial multiplication and division is to represent polynomials as linked lists of terms sorted in a term ordering and to use repeated merging. This results in poor performance on large sparse polynomials.

In this paper we introduce the balanced traveling salesman problem (BTSP), which can be used to model optimization problems where equitable distribution of resources are important. BTSP is obviously NP-hard. Efficient heuristic algorithms are presented to solve the problem along with extensive computational results using benchmark problems from TSPLIB and random instances. Our algorithms produced provably optimal solutions for several