An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Computer Theorem Proving and HoTT

Theorem-proving is a one-player game. The history of computer programs being the players goes back to 1956 and the ‘LT’ LOGIC THEORY MACHINE of Newell, Shaw and Simon. In game-playing terms, the ‘initial position’ is the core set of axioms chosen for the particular logic and the ‘moves’ are the rules of inference. Now, the Univalent Foundations Program at IAS Princeton and the resulting ‘HoTT’ book on Homotopy Type Theory have demonstrated the success of a new kind of experimental mathematics using computer theorem proving.

A proof of van Aubel's theorem using orthogonal vectors

We show how two linearly independent vectors can be used to construct two orthogonal vectors of equal magnitude in a simple way. The proof that the constructed vectors are orthogonal and of equal magnitude is a good exercise for students studying properties of scalar and vector triple products. We then show how this result can be used to prove van Aubel's theorem that relates the two line segments joining the centres of squares on opposite sides of a plane quadrilateral.

Unimodality of wave amplitude in the northern Hemisphere

A novel statistic for local wave amplitude of the 500-hPa geopotential height field is introduced. The statistic uses a Hilbert transform to define a longitudinal wave envelope and dynamical latitude weighting to define the latitudes of interest. Here it is used to detect the existence, or otherwise, of multimodality in its distribution function. The empirical distribution function for the 1960-2000 period is close to a Weibull distribution with shape parameters between 2 and 3. There is substantial interdecadal variability but no apparent local multimodality or bimodality. The zonally averaged wave amplitude, akin to the more usual wave amplitude index, is close to being normally distributed. This is consistent with the central limit theorem, which applies to the construction of the wave amplitude index. For the period 1960-70 it is found that there is apparent bimodality in this index. However, the different amplitudes are realized at different longitudes, so there is no bimodality at any single longitude. As a corollary, it is found that many commonly used statistics to detect multimodality in atmospheric fields potentially satisfy the assumptions underlying the central limit theorem and therefore can only show approximately normal distributions. The author concludes that these techniques may therefore be suboptimal to detect any multimodality.

A geostatistical analysis of the spatial variation of soil mineral nitrogen and potentially available nitrogen within an arable field.

The technology for site-specific applications of nitrogen (N) fertilizer has exposed a gap in our knowledge about the spatial variation of soil mineral N, and that which will become available during the growing season within arable fields. Spring mineral N and potentially available N were measured in an arable field together with gravimetric water content, loss on ignition, crop yield, percentages of sand, silt, and clay, and elevation to describe their spatial variation geostatistically. The areas with a larger clay content had larger values of mineral N, potentially available N, loss on ignition and gravimetric water content, and the converse was true for the areas with more sandy soil. The results suggest that the spatial relations between mineral N and loss on ignition, gravimetric water content, soil texture, elevation and crop yield, and between potentially available N and loss on ignition and silt content could be used to indicate their spatial patterns. Variable-rate nitrogen fertilizer application would be feasible in this field because of the spatial structure and the magnitude of variation of mineral N and potentially available N.

Acoustic scattering by mildly rough unbounded surfaces in three dimensions

For a nonlocally perturbed half- space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single-and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half- space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth ( Lyapunov) we show that the integral operators are nevertheless bounded as operators on L-2(Gamma) and on L-2(Gamma G) boolean AND BC(Gamma) and that the operators depend continuously in norm on the wave number and on G. We further show that for mild roughness, i.e., a surface G which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space L-2(Gamma) boolean AND BC(Gamma) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.

Beurling zeta functions, generalised primes, and fractal membranes

We study generalised prime systems P (1 < p(1) <= p(2) <= ..., with p(j) is an element of R tending to infinity) and the associated Beurling zeta function zeta p(s) = Pi(infinity)(j=1)(1 - p(j)(-s))(-1). Under appropriate assumptions, we establish various analytic properties of zeta p(s), including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of zeta p(s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N-2. Some of the above results are relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.

Force constants and dipole moment derivatives of molecules from perturbed Hartree-Fock calculations II. Applications to limited basis set SCF-MO wavefunctions

The perturbed Hartree–Fock theory developed in the preceding paper is applied to LiH, BH, and HF, using limited basis‐set SCF–MO wavefunctions derived by previous workers. The calculated values for the force constant ke and the dipole‐moment derivative μ(1) are (experimental values in parentheses): LiH, ke  =  1.618(1.026)mdyn/Å,μ(1)  =  −18.77(−2.0±0.3)D/ÅBH,ke  =  5.199(3.032)mdyn/Å,μ(1)  =  −1.03(−)D/Å;HF,ke  =  12.90(9.651)mdyn/Å,μ(1)  =  −2.15(+1.50)D/Å. The values of the force on the proton were calculated exactly and according to the Hellmann–Feynman theorem in each case, and the discrepancies show that none of the wavefunctions used are close to the Hartree–Fock limit, so that the large errors in ke and μ(1) are not surprising. However no difficulties arose in the perturbed Hartree–Fock calculation, so that the application of the theory to more accurate wavefunctions appears quite feasible.

The biocontrol fungus Pochonia chlamydosporia shows nematode host preference at the infraspecific level

A RAPD-PCR assay was developed and used to test For competitive variability in growth of the nematode biological control fungus Pochonia chlamydosporia. Saprophytic competence in soil with or without tomato plants was examined in three isolates of the fungus: RES 280 (J), originally isolated from potato cyst nematode (PCN) cysts; RES 200 (1) and RES 279 (S), both originally isolated from root knot nematode (RKN) eggs. Viable counts taken at 70 d indicated that I was the best saprophyte followed by S, with J the poorest. RAPD-PCR analysis of colonies from mixed treatments revealed that there was a cumulative effect of adding isolates to the system. This Suggested that the isolates did not interact and that they may occupy separate niches in soil and the rhizosphere. To investigate parasitic ability, soils were seeded with two isolates of the fungus: J and S, singly or in combination. Tomato or potato plants were grown in these soils; free of nematodes, or inoculated with PCN or RKN, and incubated for 77 d. The abundance of the PCN isolate J in PCN cysts was significantly greater than that of the RKN isolate S but in RKN egg masses, S was significantly more abundant than J. RAPD-PCR analysis of colonies from mixed treatments confirmed that J was more abundant than S ill PCN cysts whereas the converse was observed on RKN egg masses. This substantiates the phenomenon of nematode host preference at the infraspecific level of P. chlamydosporia and highlights its relevance for biological control of plant parasitic nematodes.

The biocontrol fungus pochonia chlamydosporia shows nematode host preference at the infraspecific level

A RAPD-PCR assay was developed and used to test For competitive variability in growth of the nematode biological control fungus Pochonia chlamydosporia. Saprophytic competence in soil with or without tomato plants was examined in three isolates of the fungus: RES 280 (J), originally isolated from potato cyst nematode (PCN) cysts; RES 200 (1) and RES 279 (S), both originally isolated from root knot nematode (RKN) eggs. Viable counts taken at 70 d indicated that I was the best saprophyte followed by S, with J the poorest. RAPD-PCR analysis of colonies from mixed treatments revealed that there was a cumulative effect of adding isolates to the system. This Suggested that the isolates did not interact and that they may occupy separate niches in soil and the rhizosphere. To investigate parasitic ability, soils were seeded with two isolates of the fungus: J and S, singly or in combination. Tomato or potato plants were grown in these soils; free of nematodes, or inoculated with PCN or RKN, and incubated for 77 d. The abundance of the PCN isolate J in PCN cysts was significantly greater than that of the RKN isolate S but in RKN egg masses, S was significantly more abundant than J. RAPD-PCR analysis of colonies from mixed treatments confirmed that J was more abundant than S ill PCN cysts whereas the converse was observed on RKN egg masses. This substantiates the phenomenon of nematode host preference at the infraspecific level of P. chlamydosporia and highlights its relevance for biological control of plant parasitic nematodes.

Diet, fecal water, and colon cancer - development of a biomarker

Colorectal cancer (CRC) is a leading cause of cancer incidence worldwide. Lifestyle factors, especially dietary intake, affect the risk of CRC development. Suitable risk biomarkers are required in order to assess the effect that specific dietary components have on CRC risk. The relationship between dietary intake and indicators of fecal water activity has been assessed using cell and animal models as well as human studies. This review summarizes the literature on fecal water and dietary components with a view to establishing further the potential role of fecal water as a source of CRC risk biomarkers. The literature indicates that fecal water activity markers are affected by specific dietary components linked with CRC risk: red meat, saturated fats, bile acids, and fatty acids are associated with an increase in fecal water toxicity, while the converse appears to be true for calcium, probiotics, and prebiotics. However, it must be acknowledged that the study of fecal water is still in its infancy and a number of issues need to be addressed before its usefulness can be truly gauged.

HAWAIIAN SKIRT: an F-box gene that regulates organ fusion and growth in arabidopsis

A fast neutron-mutagenized population of Arabidopsis ( Arabidopsis thaliana) Columbia-0 wild-type plants was screened for floral phenotypes and a novel mutant, termed hawaiian skirt ( hws), was identified that failed to shed its reproductive organs. The mutation is the consequence of a 28 bp deletion that introduces a premature amber termination codon into the open reading frame of a putative F-box protein ( At3g61590). The most striking anatomical characteristic of hws plants is seen in flowers where individual sepals are fused along the lower part of their margins. Crossing of the abscission marker, Pro(PGAZAT):beta-glucuronidase, into the mutant reveals that while floral organs are retained it is not the consequence of a failure of abscission zone cells to differentiate. Anatomical analysis indicates that the fusion of sepal margins precludes shedding even though abscission, albeit delayed, does occur. Spatial and temporal characterization, using Pro(HWS):beta-glucuronidase or Pro(HWS):green fluorescent protein fusions, has identified HWS expression to be restricted to the stele and lateral root cap, cotyledonary margins, tip of the stigma, pollen, abscission zones, and developing seeds. Comparative phenotypic analyses performed on the hws mutant, Columbia-0 wild type, and Pro(35S):HWS ectopically expressing lines has revealed that loss of HWS results in greater growth of both aerial and below-ground organs while overexpressing the gene brings about a converse effect. These observations are consistent with HWS playing an important role in regulating plant growth and development.