An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

New insights into the supression of plant pathogenic fungus (Phytophthora cinnamomi) by compost leachates

Use of compost as a soil conditioner and low-grade fertiliser is gaining popularity worldwide (Epstein, 1997). Compost not only adds plant nutrients to the soil, but also improves physical properties of soil such as buffering capacity, cation exchange capacity and water holding capacity. In addition to these benefits, compost can also suppress plant diseases caused by Phytophthora cinnamomi (Hoitink et al., 1977), Pythium aphanidermatum (Mandelbaum and Hadar, 1990), Rhizoctonia solani and Sclerotium rolfoii (Gorodecki and Hadar, 1990). Irwin et al., (1995) reported that the diseases caused by P. cinnamomi are directly responsible for considerable economic losses in many horticultural, ornamental and forestry industries throughout Australia. Phytophthora spp. continue to be the focus of attention of many researchers due to the diversity of P. cinnamomi-host interactions and their potential economic impact on a wide range of industries. The practise of using methyl bromide and other chemicals for disinfection of soil is widespread (Trill as et al., 2002). However, the use of methyl bromide and other chemicals is phased out in the USA and Europe. The suppression of soil-borne plant fungus by composts produced from tree barks (Spencer et al., 1982) and municipal solid wastes is well documented (Trill as et al., 2002). Composts that suppress plant disease have been extensively described and are used in greenhouse production systems (Lazarovitis et aI, 2001). However, most studies have focused on compo sting different types of materials and their effect on fungal pathogens inhibition rather than compo sting conditions that may produce suppressive composts. An objective of this study was to investigate the role of moisture, aeration and compost maturity in enhancing the inhibition effect of compost on the plant pathogen P. cinnamomi. A further objective was to generate an increased understanding of the mechanism of growth inhibition

Prime numbers, quantum field theory and the Goldbach conjecture

Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\dag}$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.Comment: 20 pages in A4 format, 2 figure

Flight evaluation of the STOL flare and landing during night operations

Simulated instrument approaches were made to Category 1 minimums followed by a visual landing on a 100 x 1700 ft STOL runway. Data were obtained for variations in the aircraft's flare response characteristics and control techniques and for different combinations of aircraft and runway lighting and a visual approach slope indication. With the complete aircraft and runway lighting and visual guidance no degradation in flying qualities or landing performance was observed compared to daylight operations. elimination of the touchdown zone floodlights or the aircraft landing lights led to somewhat greater pilot workload; however, the landing could still be accomplished successfully. Loss of both touchdown zone and aircraft landing lights led to a high workload situation and only a marginally adequate to inadequate landing capability

A comparison of the in vitro and in planta responses of Phytophthora cinnamomi isolates to phosphite

Research in plant pathology often relies on testing interactions between a fungicide and a pathogen in vitro and extrapolating from these results what may happen in planta. Likewise, results from glasshouse experiments are used to estimate what will happen if the fungicide is applied in the field. However, it is difficult to obtain conditions in vitro and in the glasshouse which reflect the conditions where the fungicide may eventually be used, in the field. The aim of this paper is to compare results of the effect of phosphite on P. cinnamomi isolates in vitro and in planta

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems