Bell Inequalities in Four Dimensional Phase Space and the Three Marginal Theorem

We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e. those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables qi and momentum variables pj, we establish the two following points: i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2) and (p1,p2), there does not always exist a positive phase space density rho({qi},{pj}) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-like inequalities in phase space which have their own theoretical and experimental interest. ii) given instead at most three compatible probabilities, there always exist an associated phase space density rho({qi},{pj}); the solution is not unique and its general form is worked out. These two points constitute our ``three marginal theorem''.Comment: 21 pages, Latex, no figure

Economic Analysis of Menthol Mint Cultivation in Uttar Pradesh: A Case Study of Barabanki District

The present study has been carried out in the Barabanki district of Uttar Pradesh on economic analysis of menthol mint cultivation in the year 2010. The economics has been worked out by comparing costs and returns at different stages by the conventional method. The linear production function has been fitted to evaluate the resources-use efficiency in the production of menthol mint. The study has shown that the major portion of operational cost is shared by hired labour, interculture operations, distillation charges, irrigation and machine / tractor charge. The overall benefit-cost ratio has been found to be 2.55, which indicates a higher profit for farmers on less investment in mint cultivation. The independent variables like human labour, machinery, manures and fertilizer, irrigation charges and intercultural operations have shown a positive and significant impact on the returns of mentha crop in the study area. The major problems faced by the farmers are high input cost, erratic supply of electricity, lack of adequate information, infrastructural facilities, regulated markets and energy-efficient distillation units.Menthol mint, Medicinal and aromatic plants, Mentha crop, Barabanki district, Economic analysis, Agricultural and Food Policy, Q 12, Q 18,

Effect of altitude and seabuckthorn (Hippophae rhamnoides) on soil properties in dry temperate region of Himachal Pradesh

Seabuckthorn (Hippophae rhamnoides) is an ecologically and economically important plant species used for the enhancement of soil fertility, prevention of soil erosion and production of food and medicinal products in mountain ecosystem. Altitude and landuse are the major factors which conditions the nutrient status of soil. In the study six different altitudes (3390 m, 3520 m, 3560 m, 3615 m, 3790 m and 4040 m) above sea level in dry temper-ate region of Himachal Pradesh and three land use pattern viz., (seabuckthorn forest, willow forest and wasteland) were selected for the study and their impact was investigated on soil fertility. Soil properties such (pH, organic car-bon, available macronutrients N, P, K, S and exchangeable cations Ca, Mg) were determined following the standard procedures. The values of soil available nutrients under seabuckthorn varied as pH (8.2 to 7.8), organic carbon (1.05 to 2.35 per cent) N (125 to 205 kg ha-1), P (15 to 31 kg ha-1), K (94 to 284 kg ha-1), S (28 to 53 kg ha-1), ex-changeable Ca (8.56 to 10.20 cmol (p+) kg-1) and Mg (2.3 to 3.6 cmol (p+) kg-1), respectively. The soil nutrients in seabuckthorn forest were found much higher than willow forest and wasteland, especially organic carbon and availa-ble N contents. Soil nutrients were found to increase with increasing altitude and decrease with increasing soil depth. The results concluded that Hippophae rhamnoides had significant effects (p < 0.05) on soil nutrient conditions. Hence our study indicates that seabuckthorn has a big potential for soil conservation, ecological sustainability and restoration of Himalayan ecosystem