Level sets of functions and symmetry sets of smooth surface sections

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.Comment: 15 pages, Latex, 6 grouped figures. The final version will appear in Mathematics of Surfaces. Lecture Notes in Computer Science (2005

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For non-unimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.Comment: 21 pages; v2: presentation improved, typos corrected, notational conflicts eliminated. To appear in Transformation Group

Extractable phosphorus and rice yield in an ultisol of the humid forest zone in West Africa

Little attention has been devoted to calibrating soil tests for phosphorus (P) in the field for upland rice in West Africa. The information is needed to improve fertilizer P recommendations. A field experiment was conducted in the rainy season of 1994 to establish the relationship between extractable P by Bray 1 and grain yield of four upland rice cultivars grown on an Ultisol in the forest zone of Ivory Coast. The critical limit of Bray 1 P in the soil at 90% relative rice grain yield varied from 12.5 to 15.0 mg P/kg of soil for the four cultivars tested. The P critical limit in the soil tended to be lower for the cultivars of rice earlier found to be efficient at converting applied P into grain yield

Plant phosphorus and rice yield in an Ultisol of the humid forest zone in West Africa

Little attention has been devoted to calibrating plant tissue tests for phosphorus (P) in the field for upland rice in West Africa, although information is needed for improving crop production through efficient P nutrition. A field experiment was conducted under rainfed conditions in the 1994 season to establish the relationships between plant P content and grain yield of upland rice grown on an Ultisol, having a range in extractable P, in the humid forest zone of Ivory Coast. The critical limit of P content in the whole rice plant tops at the tillering stage, at 90% relative grain yield, was found to be 2 g kg‐1 P for the four upland rice cultivars tested. Total P uptake in the biomass was significantly correlated with rice grain (r=0.81, n=20) and straw (r=0.79) yields of the cultivars

Direct and residual fertilizer phosphorus effects on yield and phosphorus efficiency of upland rice in an Ultisol

Phosphorus (P) deficiency is one of the major limiting factors for crop production in highly weathered soils in the humid zone of West Africa. Very few studies have evaluated the residual value of fertilizer P to rice in these soils. Field experiments were conducted for three years (1993–1995) to determine the response of four upland rice cultivars to fertilizer P applied at 0, 45, 90, 145 and 180 kg P ha-1 only once in 1993, and to fertilizer P residues in 1994 and 1995. The soil at the experimental site, in the humid forest zone of Ivory Coast (West Africa), was an Ultisol, low in available P. Grain yields of the rice cultivars were significantly increased by fertilizer P in 1993, and by the fertilizer P residues in 1994 and 1995 although the magnitude of response decreased with time since the fertilizer was applied. The cultivars differed in cumulative agronomic and physiological efficiencies, and the efficiencies were higher at the lower rates of P. The amounts of total P removed in three successive crops were similar for all the four rice cultivars although P harvest index was 10–12% higher in the P efficient than inefficient cultivars. The results suggest that the differences observed in the P efficiency of rice cultivars are due to differences in the internal efficiency of P

Response of upland rice to phosphorus in an ultisol in the humid forest zone of West Africa

Phosphorus (P) deficiency is one of the major limiting factor for crop production in highly weathered soils in the humid tropics. Field experiments were conducted for two years (1992 and 1993) to determine P response and efficiencies of upland rice cultivars in an Ultisol, low in available P, in the forest zone of Cote d'Ivoire. The rice cultivars tested were selected from a large number of entries tested earlier for acidity tolerance. Grain yields of the cultivars were significantly increased by P application. There was little further response in grain yield at higher rates than 60 kg P ha -1. The rice cultivars differed in agronomic and physiological P efficiencies and the efficiencies were higher at lower rates of R The rooting depths of the cultivars were increased by application of P at the lowest application rate (30 kg P ha-1).The results suggest that P fertilization of soil acidity-tolerant upland rice cultivars can significantly improve the productivity of the ultisols

Long-Term Phosphorus Fertilizer Effects on Phosphorus Uptake, Efficiency, and Recovery by Upland Rice on an Ultisol

Phosphorus (P) deficiency is a major constraint to upland rice production on highly weathered, low activity clay soils in the humid zone of West Africa. There is a paucity of information on the long-term fertilizer P effects on rice on these soils. A field experiment was conducted for six years (1993–1998) to determine the response of four upland rice cultivars to fertilizer P applied at 0, 45, 90, 135, and 180 kg P ha−1 only once in 1993, and to residual P in 1994, 1995, 1996, and 1998. The experimental site was located on an Ultisol, low in available P, in the humid forest zone of Côte d'Ivoire, West Africa. This paper discusses long-term P effects on P uptake and efficiency of upland rice cultivars. The cultivars evaluated differed in cumulative agronomic and physiological P efficiencies, and the efficiencies were higher at lower P rates for the P-efficient cultivars. The P uptake response and P harvest index were affected by P rate and its residual effect, which decreased with time after P was applied. The mean cumulative recovery of applied P in five crops of four rice cultivars varied from 5.5 to 9.4%. The results suggest that the variation in P efficiency of the cultivars is due to differences in the efficiency of utilization of P for grain production

Phosphorus response of Oryza sativa, O. glaberrima, and hybrid rice cultivars on an ultisol

Phosphorus (P) deficiency is a major constraint to upland rice production on Ultisols in the humid zone of West Africa. Integrated use of P-efficient cultivars and P nutrition is needed for enhanced sustainable productivity on these soils. This article reports on the P responsiveness of interspecific rice hybrids (crosses from Oryza sativa and O. glaberrima) along with O. sativa and O. glaberrima cultivars grown on an acidic Ultisol, low in available P. The cultivars differed in yield and P-uptake response to fresh and residual P. Two interspecific cultivars gave a linear response to P and produced the greatest grain yield under direct and residual P. The O. glaberrima cultivar CG 14 did not respond to the applied P, whereas the O. sativa cultivar was moderate in its performance. Our results show that the interspecific rice cultivars have the potential to adapt and perform well on acidic upland soil

A Unified Algebraic Approach to Classical Yang-Baxter Equation

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parak\"ahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica