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The operator ℒ generated by a linear ordinary differential expression of nth order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator ℒ, where ℒ is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator ℒ possesses a system of eigenfunctions and associated functions generating an unconditional basis in L (0, 1). Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii-Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula. [ABSTRACT FROM AUTHOR]

We consider (in general noncoercive) mixed problems in a bounded domain D in ℝ for a second-order elliptic partial differential operator A( x, Ə). It is assumed that the operator is written in divergent form in D, the boundary operator B( x, Ə) is the restriction of a linear combination of the function and its derivatives to Ə D and the boundary of D is a Lipschitz surface. We separate a closed set Y ⊂ Ə D and control the growth of solutions near Y. We prove that the pair ( A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L. The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev-Slobodetskiiĭ spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms. [ABSTRACT FROM AUTHOR]

To examine the effects of subsoil NaCl salinity in relation to water stress imposed at different growth stages, wheat was grown in a heavy texture clay soil (vertosol) under glasshouse conditions in polythene lined cylindrical PVC pots (100 cm long with 10.5 cm diameter) with very low salinity level (ECe 1.0 dS/m; ESP 1.0 and Cl 30 mg/kg soil) in top 10 cm soil (10–20 cm pot zone) and low salinity level (ECe 2.5 dS/m, ESP 5, and Cl 100 mg/kg soil) in top 10–20 cm soil (20–30 cm pot zone). The plants were exposed to three subsoil salinity levels in the 20–90 cm subsoil (30–100 cm pot zone) namely low salinity (ECe: 2.5 dS/m, ESP: 5, Cl: 100 mg/kg soil), medium salinity (ECe: 4.0 dS/m, ESP: 10, Cl: 400 mg/kg) and high salinity (ECe: 11.5 dS/m, ESP: 20, Cl: 1950 mg/kg) in the subsoil (20–90 cm soil layer: 30–100 cm pot zone). Watering of plants was withheld for 20 days commencing at either early booting or anthesis or mid grain filling, and then resumed until maturity, and these treatments were compared with no water stress. Water stress commencing at anthesis stage had the most depressing effect on grain yield and water use efficiency of wheat followed by water stress at grain filling stage and early booting stage. High subsoil salinity reduced grain yield by 39.1, 24.3%, and 13.4% respectively in plants water-stressed around anthesis, early booting, and mid grain filling compared with 36.6% in well-watered plants. There was a significant reduction in root biomass, rooting depth, water uptake and water use efficiency of wheat with increasing subsoil salinity irrespective of water regimes. Plants at high subsoil salinity had 64% of their root biomass in the top 0–30 cm soil and there was a marked reduction in subsoil water uptake. Roots also penetrated below the non-saline surface into salinised subsoil and led to attain high concentration of Na and Cl and reduced Ca/Na and K/Na ratio of flag leaf at anthesis stage. Results suggest that high subsoil salinity affects root growth and water uptake, grain yield and water use efficiency even in well water plants. Water stress at anthesis stage had the most depressing effect on wheat. [ABSTRACT FROM AUTHOR]

Conifer trees are routinely manipulated hormonally to increase flowering, branching, or adjust crown shape for production purposes. This survey of internal cytokinin levels provides a background for such treatments in Abies nordmanniana, a tree of great economic interest. Reference points in the crown and root system were sampled destructively in 4- and 6-year-old trees and analyzed for a range of cytokinins by LC-MS/MS. No seasonal patterns were detected in the root samples, and a major portion of cytokinin was in conjugated forms. Dramatic and consistent seasonal changes occurred in the crown, at levels 17–65 times higher than in the root. Predominant among crown cytokinins was ZR, except in the needles where IPR was also prominent. Within the crown, cytokinin profiles in different organs differed consistently. The leader bud showed a pronounced mid-June minimum, and a maximum later in summer. Subapical buds showed the same June minimum but peaked in mid autumn at a much lower level. Maxima in these buds were preceded by peaks in the subapical stem. Parallel patterns were observed in homologous tissues on branches.This pattern is consistent with two surges beginning in the uppermost stem tissues leading to subsequent accumulation or stimulated production within the buds. Strong differential hormonal profiles between adjacent buds with different fates agree with recent evidence of localized cytokinin production. The data suggest a reduced role of root-derived cytokinins in crown development. Practical cytokinin treatments for crown-shape regulation require close attention to dosage as well as precise timing and positioning. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas for non-self-adjoint operators generated by the Sturm-Liouville system and quasiperiodic boundary conditions. Using these asymptotic formulas, we obtain conditions on the potential for which the system of root vectors of the operator under consideration forms a Riesz basis. [ABSTRACT FROM AUTHOR]

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL2(0, 1). [ABSTRACT FROM AUTHOR]

Hairy rootlets, aggregated in longitudinal rows to form distinct clusters, are a major part of the root system in some species. These root clusters are almost universal (1600 species) in the family Proteaceae (proteoid roots), with fewer species in another seven families. There may be 10–1000 rootlets per cm length of parent root in 2–7 rows. Proteoid roots may increase the surface area by over 140× and soil volume explored by 300× that per length of an equivalent non-proteoid root. This greatly enhances exudation of carboxylates, phenolics and water, solubilisation of mineral and organic nutrients and uptake of inorganic nutrients, amino acids and water per unit root mass. Root cluster production peaks at soil nutrient levels (P, N, Fe) suboptimal for growth of the rest of the root system, and may cease when shoot mass peaks. As with other root types, root cluster production is controlled by the interplay between external and internal nutrient levels, and mediated by auxin and other hormones to which the process is particularly sensitive. Proteoid roots are concentrated in the humus-rich surface soil horizons, by 800× in Banksia scrub-heath. Compared with an equal mass of the B horizon, the A1 horizon has much higher levels of N, P, K and Ca in soils where species with proteoid root clusters are prominent, and the concentration of root clusters in that region ensures that uptake is optimal where supply is maximal. Both proteoid and non-proteoid root growth are promoted wherever the humus-rich layer is located in the soil profile, with 4× more proteoid roots per root length in Hakea laurina. Proteoid root production near the soil surface is favoured among hakeas, even in uniform soil, but to a lesser extent, while addition of dilute N or P solutions in split-root system studies promotes non-proteoid, but inhibits proteoid, root production. Local or seasonal applications of water to hakeas initiate non-proteoid, then proteoid, root production, while waterlogging inhibits non-proteoid, but promotes proteoid, root production near the soil surface. A chemical stimulus, probably of bacterial origin, may be associated with root cluster initiation, but most experiments have alternative interpretations. It is possible that the bacterial component of soil pockets rich in organic matter, rather than their nutrient component, could be responsible for the proliferation of proteoid roots there, but much more research on root cluster microbiology is needed. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the completeness of the root function system for nonlocal elliptic problems in the Sobolev spaces Wp l(Q). [ABSTRACT FROM AUTHOR]

Indices of shallow rootedness and fractal methods of root system study were combined with sapflow monitoring to determine whether these 'short-cut' methods could be used to predict tree competition with crops and complementarity of below ground resource use in an agroforestry trial in semi-arid Kenya. These methods were applied to Grevillea robusta Cunn., Gliricidia sepium (Jacq.) Walp., Melia volkensii Gurke and Senna spectabilis syn. Cassia spectabilis aged two and four yearswhich were grown in simultaneous linear agroforestry plots with maize as the crop species. Indices of competition (shallow rootedness) differed substantially according to tree age and did not accurately predict tree:crop competition in plots containing trees aged four years.Predicted competition by trees on crops was improved by multiplying the sum of proximal diameters squared for shallow roots by diameter at breast height^2, thus taking tree size into account. Fractal methods for the quantification of total length of tree root systems worked well with the permanent structural root system of trees but seriouslyunderestimated the length of fine roots (less than 2 mm diameter). Sap flow measurements of individual roots showed that as expected, deep tap roots provided most of the water used by the trees during the dry season. Following rainfall, substantial water uptake by shallow lateral roots occurred more or less immediately, suggesting that existing roots were functioning in the recently wetted soil and that there was no need for new fine roots to be produced to enable water uptake following rainfall. [ABSTRACT FROM AUTHOR]

This study was designed to investigate the strength of attachment ofplant seedling roots to the soil in which they were grown. The studyalso assessed the effects of differing soil textures and differing soil matric potentials upon the strength of the root:soil attachment. A device for growing roots upon a soil surface was designed, and was used to produce roots which were attached to the soil. In order to quantify root:soil adhesion, roots of maize seedlings, grown on the soil surface, were subsequently peeled off using a universal test machine, in conjunction with simultaneous time-lapse video observation. To clarify the partitioning of energy in the root:soil peeling test, separate mechanical tests on roots, and on two adherent remoulded topsoil balls were also carried out. The seedling root was characterised bya low bending stiffness. The energy stored in bending was negligible, compared to the root:soil adhesion energy. The mechanical properties of two adherent remoulded topsoil balls were a decrease of the soil:soil adhesion energy as the soil:soil plastic energy increased. These two parameters were therefore interdependent. Using a video-camera system, it was possible to separate the different processes occurringduring the root:soil peeling test, in particular, the seed:soil adhesion and the root:soil soil adhesion. An interpretation of the complex and variable force:displacement curves was thus possible, enabling calculation of the root:soil interfacial rupture energy. At a given suction (10 kPa), the results of the peeling test showed a clear soil texture effect on the value of the root:soil interfacial rupture energy. In contrast, for the same silty topsoil, the effect of the soil water suction on the value of the interfacial rupture energy was very moderate. The root:soil interfacial rupture energy was controlled mainly by a product of microscopic soil specific surface area and the macroscopic contact surface area between the root and the soil. Biological [ABSTRACT FROM AUTHOR]

Phosphorus deficiency-induced metabolic changes related to exudationof carboxylic acids and protons were compared in roots of wheat (Triticum aestivum L. cv Haro), tomato (Lycopersicon esculentum L., cv. Moneymaker), chickpea (Cicer arietinum) and white lupin (Lupinus albusL. cv. Amiga), grown in a hydroponic culture system. P deficiency strongly increased the net release of protons from roots of tomato, chickpea and white lupin, but only small effects were observed in wheat.Release of protons coincided with increased exudation of carboxylic acids in roots of chickpea and white lupin, but not in those of tomato and wheat. P deficiency-induced exudation of carboxylic acids in chickpea and white lupin was associated with a larger increase of carboxylic acid concentrations in the roots and lower accumulation of carboxylates in the shoot tissue compared to that in wheat and tomato. Citric acid was one of the major organic acids accumulated in the rootsof all investigated species in response to P deficiency, and this was associated with increased activity and enzyme protein levels of PEPcarboxylase, which is required for biosynthesis of citrate. Accumulation of citric acid was most pronounced in the roots of P-deficient white lupin, chickpea and tomato. Increased PEP carboxylase activity in the roots of these plants coincided with decreased activity of aconitase, which is involved in the breakdown of citric acid in the TCA cycle. In the roots of P-deficient wheat plants, however, the activities of both PEP carboxylase and aconitase were enhanced, which was associated with little accumulation of citric acid. The results suggest that P deficiency-induced exudation of carboxylic acids depends on the ability to accumulate carboxylic acids in the root tissue, which inturn is determined by biosynthesis, degradation and partitioning of carboxylic acids or related precursors between roots and shoot. In some plant species such as white lupin, there are indications for a specific [ABSTRACT FROM AUTHOR]

Root mucilage was collected from 3-4 day-old axenically-grown maize seedlings (Zea mays L. cv. Freya). The water potential of the hydrated mucilage was measured by thermocouple psychrometry and the rheologyat low deformation rates was studied using an oscillating cone and plate rheometer which provides information on both the elastic and viscous components of its behaviour. Water potential decreased as mucilage solute concentration increased, reaching a value of ?60kPa at 1.2 mg mL-1. At the lowest oscillation rate, the mucilage had a dynamic viscosity of 145 mPa s and behaved as a weak viscoelastic gel. After filtration to remove suspended root cap cells and other solid plant material, mucilage viscosity was reduced to 5-10 mPa s at low oscillation rates and the behaviour was that of a viscous liquid. The decrease in viscosity which occurs on filtration indicates that the root cap cells form an integral part of the gel system, either by interacting directly with each other or via the polysaccharide. Our observations provide further support for the idea that mucilage plays amajor role in maintaining root-soil contact in the rhizosphere. [ABSTRACT FROM AUTHOR]

A mathematical model of P solubilization by organic anion excretion from roots is described and used to account for P solubilization by rice (Oryza sativa L.) plants growing in aerobic soil. In previous experiments with rice in an aerobic, highly-weathered, P-deficient soil,we found that the plants were able to solubilize P from an alkali-soluble pool and thereby increase their P uptake. The solubilization could not be explained by pH changes nor by the release of phosphatases. In subsequent experiments we found excretion of citrate from rice roots into nutrient solutions, and the synthesis and excretion of citrate tended to increase under P starvation. The model allows for the diffusion of the organic anion away from a root, its decomposition by soil microbes, its reaction with the soil in solubilizing P, and diffusion of the solubilized P back towards the root as well as away fromit. We calculated the rate of citrate excretion from rice roots growing in soil based on measured steady-state citrate concentrations in the rhizosphere and calculated rates of decomposition. Calculations using these and other model parameters obtained independently showed that the observed solubilization and increased P uptake by rice growing in soil could be accounted for. A sensitivity analysis of the modelis given. [ABSTRACT FROM AUTHOR]

Incorporation of cover crops into cropping systems may contribute toa more efficient utilization of soil and fertilizer P by less P-efficient crops through exudation of P-mobilizing compounds by the roots of P-efficient plant species. The main objective of the present work was to test this hypothesis. First a method has been developed which allows the quantification of organic anion exudation from individual cluster roots formed by P-deficient white lupin (Lupinus albus L.). Lupin plants were grown in nutrient solution at 1 mu M P and in a low P loess in small rhizotrons. Organic anions exuded from intact plantsgrown in nutrient solution were collected from individual cluster roots and root tips sealed in small compartments by an anion-exchange resin placed in nylon bags (resin-bags). Succinate was the dominant organic anion exuded followed by citrate and malate. The mean of citrate exudation-rate was 0.06 pmol mm-1 s-1 with exudation highly dependent on the citrate concentration and on the ageof the cluster roots. Exudates from cluster roots and root tips grown at the soil surface (rhizotron-grown plants) were collected using overlayered resin-agar (resin mixed with agar). Citrate exudation fromcluster roots was 10 times higher than that from root tips. Fractionation of P in the cluster root rhizosphere-soil indicates that white lupin can mobilize P not only from the available and acid-soluble P, but also from the stable residual soil P fractions. In pot experiments with an acid luvisol derived from loess low in available P, growth of wheat was significantly improved when mixed-cropped with white lupin due to improved P uptake. Both in mixed culture and in rotation wheat could benefit from the P mobilization capacity of white lupin, supporting the hypothesis above. Nine tropical leguminous cover crops and maize were grown in a pot experiment using a luvisol from NorthernNigeria low in available P. All plant species derived most of thei rP fro [ABSTRACT FROM AUTHOR]

Plants require roots to supply water, nutrients and oxygen for growth. The spatial distribution of roots in relation to the macropore structure of the soil in which they are growing influences how effectivethey are at accessing these resources. A method for quantifying root-macropore associations from horizontal soil sections is illustrated using two black vertisols from the Darling Downs, Queensland, Australia. Two-dimensional digital images were obtained of the macropore structure and root distribution for an area 55 x 55 mm at a resolution of 64 mu m. The spatial distribution of roots was quantified over a range of distances using the K-function. In all specimens, roots were shown to be clustered at short distances (1-10 mm) becoming more random at longer distances. Root location in relation to macropores was estimated using the function describing the distance of each root to the nearest macropore. From this function, a summary variable, termed the macropore sheath, was defined. The macropore sheath is the distance from macropores within which 80% of roots are located. Measured root locations were compared to random simulations of root distributionto establish if there was a preferential association between roots and macropores. More roots were found in and around macropores than expected at random. [ABSTRACT FROM AUTHOR]

Under certain soil conditions, e.g. hardsetting clay B-horizons of South-Eastern Australia, wheat plants do not perform as well as would be expected given measurements of bulk soil attributes. In such soils, measurement indicates that a large proportion (80%) of roots are preferentially located in the soil within 1 mm of macropores. This paper addresses the question of whether there are biological and soil chemical effects concomitant with this observed spatial relationship. The properties of soil manually dissected from the 1-3 mm wide region surrounding macropores, the macropore sheath, were compared to those that are measured in a conventional manner on the bulk soil. Field specimens of two different soil materials were dissected to examine biological differentiation. To ascertain whether the macropore sheath soil differs from rhizosphere soil, wheat was grown in structured and repacked cores under laboratory conditions. The macropore sheath soil contained more microbial biomass per unit mass than both the bulk soiland the rhizosphere. The bacterial population in the macropore sheath was able to utilise a wider range of carbon substrates and to a greater extent than the bacterial population in the corresponding bulk soil. These differences between the macropore sheath and bulk soil were almost non-existent in the repacked cores. Evidence for larger numbers of propagules of the broad host range fungus Pythium in the macropore sheath soil were also obtained. [ABSTRACT FROM AUTHOR]