On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.Comment: 4 figures. Added some comments about total curvature and other detail

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Not AvailableA field experiment was conducted at the Research Farm of ICAR-Indian Institute of Soil Science, Bhopal, India to study the effect of different combinations of organic inputs on performance of soybean [Glycine max (L.) Merr.]. Soybean (cv. JS 335) was grown in a randomized block design with seven treatments and four replications. The treatments studied were organic manure (OM) (T1); biodynamic preparation (BD) (T2); OM+panchagavya (OM+PG) (T3); OM+BD (T4); OM+PG+BD (T5); control (T6); and recommended dose of fertilizers (RDF) (T7). The results of the study revealed that, the treatments with organic manures either alone or in combination with panchagavya and/or biodynamic application improved the performance of soybean crop. The yield of soybean was found comparable and even better (higher 5–13%) with those obtained under RDF, which enhanced nutrient removal by soybean crop by 5–13% with organic combinations. Similarly, the application of organic manures significantly improved the available nutrient status of soil and showed 19–22% and 28–33% higher available N and P, respectively than RDF. Soil enzyme activities, viz. dehydrogenase and alkaline phosphatase increased by 62–72% and 27–35%, respectively under the treatments receiving organic source of nutrients over RDNot Availabl