101 results for Report, 1997

Recent Developments in Organic Food Production in New Zealand: Part 2: Kiwifruit in the Bay of Plenty
Campbell, Hugh; Fairweather, John; Steven, David (1997)
Report
University of OtagoThis report presents the findings of research into the development of organic kiwifruit production in the Bay of Plenty. These results form the second of four case studies which constitute the Public Good Science Fund programme ‘Optimum Development of Certified Organic Horticulture in New Zealand’. The other case study regions are Canterbury (Campbell 1996), Gisborne (to be completed during 1997) and Nelson (to be completed by 1998). The primary objective of this report is to document developments in the organic export industry in the Bay of Plenty. Comparisons between Canterbury and the Bay of Plenty have occasionally been included in this report in order to provide more clarity about the development of organic production in the Bay of Plenty itself. While there is some discussion of the differences between Canterbury and the Bay of Plenty in the Conclusion, these are only brief. Full comparison of the regional factors influencing the development of organic exporting will be set aside until all four case studies have been completed.
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A Constructive Proof of Gleason's Theorem
Richman, F; Bridges, D.S (199705)
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The University of Auckland LibraryGleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. In this paper we give a constructive proof of that theorem. A measure μ on the projections of a real or complex Hilbert space assigns to each projection P a nonnegative real number μ(P) such that if σ = ∑Pi, where the Pi are mutually orthogonal, then μ(σ) =∑μ(Pi). Such a measure is determined by its values on the onedimensional projections. Let W be the measure of the identity projection, and Px the projection onto the 1dimensional space spanned by the unit vector x. Then the measure μ is determined by the realvalued function f(x) = μ(Px) on the unit sphere, a function which has the property that [see pdf for formula] for each orthonormal basis E. Gleason calls such a function f a frame function of weight W. If T is a positive operator of trace class, then f(x) = ‹Tx,x› is a frame function. Gleason's theorem is that every frame function arises in this way. The original reference for Gleason's theorem is [4], which can also be found in Hooker [6]. Cooke, Keane and Moran [3] gave a proof that is elementary in the sense that it does not appeal to the theory of representations of the orthogonal group, which the original proof does. However, some of the reasoning in [3] seems hopelessly nonconstructive, so we follow the general outline of [4] until we come to the end of the 3dimensional real case, at which point we modify some arguments in [3] rather than attempt a constructive development of the necessary representation theory. Any Hermitian form B on a finitedimensional inner product space gives rise to a frame function f(x) = B(x; x) whose weight is equal to the trace of the matrix of B. The essence of Gleason's theorem is the following converse.  from Introduction
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A Genius' Story: Two Books on Godel
Calude, C.S (199706)
Report
The University of Auckland LibraryUndoubtly, Gödel was the greatest logician of the twentieth century. There is no trace of exaggeration in saying, following Wang, that Gödel's contribution to mathematics has the same status as Freudian psychology, Einstein's theory of relativity, Bohr's principle of complementarity, Heisenberg's uncertainty principle, keynesian economics, and Watson and Crick double helix model of DNA. Yet, with a few notable exceptions, most of the personal details of Gödel's life remained a mystery
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Clusters of Two Player Games and Restricted Determinacy Theorem
Khoussainov, B; Yakhnis, A; Yakhnis, V (199705)
Report
The University of Auckland LibraryWe introduce a new notion of a cluster of infinite two player games between players 0 and 1. This is an infinite collection of games whose game trees can be composed into a graph which is similar to a tree except that the graph might not have the initial node. For each node of the graph there is an ancesstor node. We call this graph the arena of the cluster. For a game cluster we introduce a notion of a winner for the whole cluster. This notion is weaker than the requirement to win every game of the cluster. Any two player game can be viewed as a game cluster consisting of all its residual games [3, 18]. We extend the restricted memory determinacy (RMD) theorem of GurevichHarrington (GH), [3] to game clusters. We think that the notion of a game cluster improves the modeling power of two player games used to give semantics for concurrent processes [10, 11].
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Imprecise Reasoning about Geographic Information
Guesgen, H.W (199709)
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The University of Auckland LibraryThis report summarizes preliminary work on a framework for imprecise reasoning about spatial information, in particular spatial information in geographic information systems. It is based on papers previously published by Jonathan Histed, Ute Lörch, David Poon, and the author. Geographic information systems have gained an increasing interest over the recent years. However, their abilities are restricted in that they usually reason about precise quantitative information only, which means that they fail whenever exact matches cannot be found. They do not allow for any form of reasoning with imprecision. In this report, we describe a way of incorporating imprecise qualitative spatial reasoning with quantitative reasoning in geographic information systems. In particular, we show how tessellation data models can be extended to allow for qualitative spatial reasoning. The idea is to associate qualitative information with fuzzy sets whose membership grades are computed by applying the concept of proximity. In addition, we will show how images like geographic maps or satellite images can be analyzed by computing the distances between given reference colors and the colors that occur in the image, and how the results of this analysis can be used in the fuzzy spatial reasoner.
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DNA Computing Based on Splicing: Universality Results
Paun, G (199712)
Report
The University of Auckland LibraryFirst, we recall some characterizations of recursively enumerable languages by means of finite H systems with certain regulations on the splicing operation. Then, we consider a variant of the splicing operation where the splicing proceeds always in couples of steps: the two strings obtained after a splicing enter immediately a second splicing (the rules used in the two steps are not prescribed). Somewhat surprising if we take into account the loose control on the performed operations, extended H systems with finite sets of axioms and of splicing rules, using this double splicing operation, can again characterize the recursively enumerable languages. Finally, we consider two types of distributed H systems: communicating distributed H systems and timevarying distributed H systems. For the first type of devices, we give a new proof of the recent result of [24] that (in the extended case) such systems with three components characterize the recursively enumerable languages. In what concerns the second mentioned distributed model, we prove that timevarying H systems with seven components can characterize the recursively enumerable languages. The optimality of these two last mentioned results is open.
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Effective Presentability of Boolean Algebras of CantorBendixson Rank 1
Downey, R.G; Jockusch Jr, C.G (199708)
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The University of Auckland LibraryWe show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B=I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite CantorBendixson rank.
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Cooperating Distributed Splicing Systems
MartinVide, C; Paun, G (199712)
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The University of Auckland LibraryWe introduce a new class of cooperating distributed H systems which consist of a given set of splicing systems (sets of splicing rules plus sets of axioms), similar in form to the cooperating distributed grammar systems. By applying iteratively the components of such a system (starting from a given initial string), in a sequence which runs nondeterministically, in such a way that a step is considered correctly finished only if no more splicing is possible, we obtain a language. Somewhat surprisingly if we take into account the loose control on the operations we carry out, a characterization of recursively enumerable languages is obtained, by mechanisms as above with only three components. We also characterize the recursively enumerable languages by cooperating distributed H systems with the components containing at most three splicing rules (in this case the number of components is no longer bounded).
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Deterministic Incomplete Automata: Simulation, Universality and Complementarity
Calude, E; Lipponen, M (199706)
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The University of Auckland LibraryWe study finite deterministic incomplete automata without initial states. This means that at any stage of a computation there is at most one transition to the next state. We will first investigate how two incomplete automata can simulate each other. Further on we construct an incomplete automaton which simulates a given automaton S and has the minimum number of states compared to any other automaton simulating S. Finally, we study Moore's uncertainty principles for incomplete automata. In contrast with the case of complete automata, it is possible to construct incomplete threestate automata displaying both types of complementarity.
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The Effective Riemann Mapping Theorem
Hertling, P (199711)
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The University of Auckland LibraryThe main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping theorem by Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which topological information precisely one needs about a nonempty, proper, open, connected, and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from this set onto the unit disk, and vice versa, which topological information about the set can be obtained from a description of a holomorphic bijection. The second version, which is derived from the first by considering the sets and the functions with computable descriptions, characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection onto the unit disk. This solves a problem posed by PourEl and Richards. We also show that this class of sets is strictly larger than a class of sets considered by Zhou, which solves an open problem posed by him. In preparation, recursively enumerable open subsets and closed subsets of Euclidean spaces are considered and several effective results in complex analysis are proved.
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Adjoints, Absolute Values and Polar Decompostions
Bridges, D.S; Richman, F; Schuster, P (199711)
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The University of Auckland LibraryVarious questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Riesz representation theorem is proved without countable choice.
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Small Trivalent Graphs of Large Girth
Conder, M (199706)
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The University of Auckland LibraryDefinitions are given for seven trivalent Cayley graphs, of girths 17; 18; 20; 21; 22; 23 and 24. At the time of writing (June 1997) each of these is the smallest known trivalent graph of the corresponding girth.
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An Explicit Construction of a Universal Extended H System
Alford, G (199708)
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The University of Auckland LibraryLately there has been much interest concerning H systems, a generative mechanism based on the splicing operation, itself a languagetheoretic equivalent of DNA recombination. Păun et al. have shown that regular extended H systems are theoretically universal but one has not yet been explicitly constructed. In this paper we explicitly construct a universal extended H system containing 182 axioms and 270 groups of rules.
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On Initial Segments of Computable Linear Orders
Coles, R.J; Downey, R.G; Khoussainov, B (199703)
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The University of Auckland LibraryWe show there is a computable linear order with a π02 initial segment that is not isomorphic to any computable linear order.
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Games with Unknown Past
Khoussainov, B; Yakhnis, A; Yakhnis, V (199708)
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The University of Auckland LibraryWe define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by GurevichHarrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends further the Gurevich Harrington (GH) determinacy theorem from [5].
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Minimal Deterministic Incomplete Automata
Calude, E; Lipponen, M (199710)
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The University of Auckland LibraryWe construct a minimal automaton for an outputincomplete Moore automa ton. The approach is motivated by physical interpretation of seeing deterministic finite automata as models for elementary particles. When compared to some classical methods our minimal automaton is unique up to an isomorphism and preserves also the undefined or unspecified behaviour of the original automaton.
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The Constructive Implicit Function Theorem and Applications in Mechanics
Bridges, D.S; Calude, C; Pavlov, B; Stefanescu, D (199711)
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The University of Auckland LibraryWe examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit Function Theorem in classical mechanics
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Finite Nondeterministic Automata: Simulation and Minimality
Calude, C.S; Calude, E; Khoussainov, B (1997 09)
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The University of Auckland LibraryMotivated by recent applications of finite automata to theoretical physics, we study the minimization problem for nondeterministic automata (with outputs, but no initial states). We use EhrenfeuchtFraïsselike games to model automata responses and simulations. The minimal automaton is constructed and, in contrast with the classical case, proved to be unique up to an isomorphism. Finally, we investigate the partial ordering induced by automata simulations. For example, we prove that, with respect to this ordering, the class of deterministic automata forms an ideal in the class of all automata.
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Practical Enumeration Methods for Graphs of Bounded Pathwidth and Treewidth
Dinneen, Michael (199709)
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The University of Auckland LibraryUsing an algebraic representation for graphs of bounded pathwidth or treewidth we provide simple methods for generating these families in increasing order of the number of vertices and edges. We also study canonic representions of fixed and free boundaried graphs of bounded width.
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Chaitin Omega Numbers and Strong Reducibilities
Calude, C.S; Nies, A (199710)
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The University of Auckland LibraryWe prove that any Chaitin Ω number (i.e., the halting probability of a universal selfdelimiting Turing machine) is wttcomplete, but not ttcomplete. In this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets. The proof is direct and elementary.
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