Point and Lie Bäcklund symmetries of certain partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of MA in Mathematics at Massey University

The aim of this thesis is to: (1) Explore the use of differential forms in obtaining point and contact symmetries of particular partial differential equations (PDEs) and hence their corresponding similarity solutions. [1] and [4]. (2) Explore the generalized or Lie-Bäcklund symmetries of particular PDEs with particular reference to the Korteweg-de Vries-Burgers (KdVB) equation [3]. Finding point symmetries of a PDE H = 0 with independent variables (x1,x2 ) which we take to represent space and time and dependent variable (u) means finding the transformation group that takes the variables (x1, x2, u) to the system (x´1, x´2 , u´ ) and maps solutions of H = 0 into solutions of the same equation. The form of H = 0 remains invariant. The transformation group is usually expressed in terms of its infinitesimal generator (X) where using the tensor summation convention. X can be considered as a differential vector operator with components (ξ1 , ξ2 , η) operating in a three dimensional manifold (space) with coordinates (x1 , x2 , u). The invariance of H = 0 under the transformation group is expressed in terms of a suitable prolongation or extension of X (denoted by X(pr) ) to cover the effect of the transformations on the derivatives of u in H = 0. The invariance condition for H = 0 under the action of the transformation group is (Pr) [H] = 0 whenever H = 0. We consider x1 , x2 , u and the derivatives of u to be independent variables. In practical terms, finding point symmetries of H = 0 means finding the components (ξ1 , ξ2 , η) of the infinitesimal generator (X). There are two general methods for finding ξ1 , ξ2 η. [From Introduction] [NB: Mathematical/chemical formulae or equations have been omitted from the abstract due to website limitations. Please read the full text PDF file for a complete abstract.

Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.Comment: 24 page

Observations on the Biological Control Agents of the American Plum Borer (Lepidoptera: Pyralidae) In Michigan Cherry and Plum Orchards

The American plum borer, Euzophera semifuneralis (Walker) (Lepidoptera: Pyralidae), is an important pest in orchards, yet little is known regarding its biological control. We performed a comprehensive survey of the natural enemy complex contributing to American plum borer control in Michigan plum and cherry orchards, while also exploring the relationship between pest infestation and tree wounding from mechanical harvesting. We scouted 30 orchards with varying degrees of tree wounding to document extent of infestations of American plum borer and another pest, the lesser peach borer, Synanthedon pictipes (Grote and Robinson) (Lepidoptera: Sessiidae). We simultaneously recorded biological control agents, including the presence of a Hirsutella fungal pathogen. Live American plum borer larvae and pupae were collected for rearing and identifying hymenopteran parasitoids. American plum borer infestations were highest in orchards with high levels of tree wounding, or in orchards that used minimum pesticides or were abandoned. Numerous organisms were documented as biological control agents including various species of birds, spiders, beetles, and ants. Ichneumon wasps were the dominant parasitoids, of which Venturia nigricoxalis (Cushman) (Hymenoptera: Ichneumonidae) was the most common. Liotryphon variatipes (Provancher) (Hymenoptera: Ichneumonidae) was com- monly reared from a closely associated sessiid pest, but not from American plum borer. Hirsutella was commonly found and had a density-dependent relation- ship with American plum borer infestations. Our information gathered on the natural enemy complex of E. semifuneralis includes many new host associations and can serve as a starting point for developing biological control programs for fruit orchards in the Great Lakes region

Household energy expenditures, 1982–2005

While energy's share of total expenditures has risen in recent years, it remains below the shares seen in the early and mid-1980s. Furthermore, the impact of the price increases on a household differs, based on the household's specific energy consumption patterns.Energy consumption ; Households - Economic aspects

Game-theoretical control with continuous action sets

Motivated by the recent applications of game-theoretical learning techniques to the design of distributed control systems, we study a class of control problems that can be formulated as potential games with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems. The method employed is to analyse the learning process under study through a mean-field dynamical system that evolves in an infinite-dimensional function space (the space of probability distributions over the players' continuous controls). To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge to equilibrium in the case of potential games. These results combine to give a provably-convergent learning algorithm in which players do not need to keep track of the controls selected by the other agents.Comment: 19 page

From the Desktop to the Cloud: Leveraging Hybrid Storage Architectures in Your Repository

4th International Conference on Open RepositoriesThis presentation was part of the session : Conference PresentationsDate: 2009-05-19 01:00 PM – 02:30 PMRepositories collect and manage data holdings using a storage device. Mainly this has been a local file system, but recently attempts have been made at using open storage products and cloud storage solutions, such as Sun's Honeycomb and Amazon S3 respectively. Each of these solutions has their own pros and cons but There are advantages in adopting a hybrid model for repository storage, combining the relative strengths of each one in a policy-determined model. In this paper we present an implementation of a repository storage layer which can dynamically handle and manage a hybrid storage systemJoint Information Systems Committee (JISC

The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2

A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class parity-P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.Comment: minor change