Phase Space Topology and Bifurcation of Liouville Torii in the Goryatchev-Tchaplygin Top

The classical problem of a rigid body with a fixed point is considered in the case of Goryatchev-Tchaplygin. We give a complete description of its real phase space topology. All generic bifurcation of Liouville Torii is determined theoretically and numerically. We give also explicit periodic solutions of the problem.The classical problem of a rigid body with a fixed point is considered in the case of Goryatchev-Tchaplygin. We give a complete description of its real phase space topology. All generic bifurcation of Liouville Torii is determined theoretically and numerically. We give also explicit periodic solutions of the problem

The response of nutrient uptake, photosynthesis and yield of tomato to biochar addition under reduced nitrogen application

Abstract Tomato is an important economic crop that is widely consumed worldwide. Tomato production is mainly limited by the use of nitrogen fertilizer, sunlight, soil and water conditions. Biochar is one of the soil amendments, and it is recognized as a promising practice for improving crop production in agriculture. The effect of biochar on the photosynthetic traits and tomato yield under reduced nitrogen fertilizer application is still not well understood. The objective of this research is to investigate the influence of biochar application on the photosynthesis and yield of tomato under reduced nitrogen fertilizer application from the perspectives of the nutrient uptake of plants (nitrogen and phosphorus), leaf photosynthetic pigment and leaf gas exchange parameters. Two-year greenhouse experiments containing six biochar levels (0, 10, 30, 50, 70, and 90 t ha⁻¹) and two nitrogen fertilizer application rates (190 and 250 kg ha⁻¹) were conducted. Compared with C0, C50 significantly improved the nitrogen uptake (74–80%) and phosphorus uptake (76–95%) by tomato plants and further enhanced the photosynthetic traits of tomato leaves (net photosynthetic rate (Pn), stomatal conductance (gs), transpiration rate (Tr) and chlorophyll (2–60%), which lead to the highest gains in tomato yield (more than 50%) even when the applied nitrogen fertilizer was significantly reduced (from 250 kg ha⁻¹ to 190 kg ha⁻¹). The photosynthesis rate had a linear correlation with the total nitrogen and phosphorus accumulation and tomato yield. The results will enhance our understandings about the effect of biochar on the photosynthesis and yield of tomato and be of importance for practical agricultural management

Biochar improves soil-tomato plant, tomato production, and economic benefits under reduced nitrogen application in northwestern China

Abstract The tomato is an important economic crop that is a main ingredient of some prepared food as well as a focus of the agricultural industry. Optimizing nitrogen (N) fertilizers is essential for sustainable agricultural development, while the excessive use of N fertilizers leads to environmental and food production problems. As a soil amendment, biochar has been widely used to improve soil quality and crop yield. However, little information is available on the effects of biochar and N fertilizer reduction on tomato plant, soil characteristics in tomato cultivation and tomato production. In this study, a greenhouse experiment was carried out in Yangling, Shaanxi province, China, including four biochar levels (0, 30, 50, and 70 t ha⁻¹) under drip irrigation and four N application rates (170, 190, 210, and 250 kg ha⁻¹). The results showed that adding too much biochar (e.g., 70 t ha⁻¹) and reducing N fertilizer too far (e.g., by 32%) will not lead to satisfactory results in terms of tomato growth, tomato yield and quality, and economic benefits. Biochar addition could significantly enhance microbial abundance, enzyme activity, and tomato growth compared with non‒biochar treatments when reducing the amount of applied N fertilizer by 16% or 24% (N2 and N3). From the perspectives of tomato yield, tomato quality (sugar‒acid ratio and vitamin C (VC) content), and economic benefits, optimal application rate of biochar and N fertilizer based on the silty clay loam soil of northwest China under drip irrigation is proposed, respectively. The proposal is based on both multidimensional nonlinear regression models and a comparison with experimental treatments. For example, biochar addition at 50 t ha⁻¹ and reducing N fertilizer by 24% achieved the greatest tomato yield. Compared with non-biochar treatment under the corresponding N fertilizer level, soil enzyme activity (urease, phosphatase, and catalase), microbial abundance (bacteria, fungi, and actinomycetes), leaf gas exchange parameters (gs, Pn, and Tr), and biomass increased on average by 88.76%, 7.49%, 43.23%, and 39.67%, respectively. Based on a comprehensive consideration of tomato yield, VC content, sugar‒acid ratio, and economic benefits, 35 t ha⁻¹ biochar and 200 kg ha⁻¹ N fertilizer is the recommended combination of biochar and nitrogen fertilizer for local farmers

Spatial price discrimination in the unidirectional Hotelling model with elastic demand

Unidirectional Hotelling model, Location equilibrium, Bertrand, Cournot, D43, L11, L41,