Some Aspects of Mathematical Programming in Statictics

The Almighty has created the Universe and things present in it with an order and proper positions and the creation looks unique and perfect. No one can even think much better or imagine to optimize these further. People inspired by these optimum results started thinking about usage of optimization techniques for solving their real life problems. The concept of constraint optimization came into being after World War II and its use spread vastly in all fields. However, in this process, still lots of efforts are needed to uncover the mysteries and unanswered questions, one of the questions always remains live that whether there can be a single method that can solve all types of nonlinear programming problems like Simplex Method solves linear programming problems. In the present thesis, we have tried to proceed in this direction and provided some contributions towards this area. The present thesis has been divided into five chapters, chapter wise summary is given below: Chapter-1 is an introductory one and provides genesis of the Mathematical Programming Problems and its use in Statistics. Relationship of mathematical programming with other statistical measures are also reviewed. Definitions and other pre-requisites are also presented in this chapter. The relevant literature on the topic has been surveyed. Chapter-2 deals with the two dimensional non-linear programming problems. We develop a method that can solve approximately all type of two dimensional nonlinear programming problems of certain class. The method has been illustrated with numerical examples. Chapter-3 is devoted to the study of n-dimensional non-linear programming problems of certain types. We provide a new method based on regression analysis and statistical distributions. The method can solve n-dimensional non-linear programming problems making use of regression analysis/co-efficient of determination. In chapter-4 we introduce a filtration method of mathematical programming. This method divides the constraints into active and non active and try to eliminate the less important constraints (non-active constraints) and solve the problem with only active constraints. This helps to find solution in less iterations and less in time while retaining optimality of the solution. The final chapter-5 deals with an interesting relationship between linear and nonlinear programming problems. Using this relationship, we can solve linear programming problems with the help of non-linear programming problems. This relationship also helps to find a better alternate solutions to the linear programming problems. In the end, a complete bibliography is provided

Cause and Post Harvest Management of Bacterial Soft Rot of Cabbage (Brassica oleracea var capitata L.)

Soft rot is one of the most destructive diseases of vegetable crops including cabbage. The disease can be found on crops in the field, transit, storage or during marketing resulting in major economic losses. Present studies aimed at studying the status, cause and its post harvest management strategy. Periodical surveys conducted during 2008-09 revealed that the soft rot of cabbage is prevalent in all the vegetable markets of Kashmir valley throughout the year with varied incidence and intensity during different months of the year. Highest average soft rot incidence of 31.54 per cent with intensity of 15.06 per cent was recorded during the month of June followed by an incidence and intensity of 29.79 and 14.70 per cent, respectively during the month of August. Among the various districts surveyed, the highest disease incidence (17.53%) and intensity (9.67%) was recorded in Anantnag and the lowest 12.87 and 6.30%, respectively in district Srinagar. The causal pathogen responsible for soft rot in cabbage under temperate conditions of Kashmir valley was identified as Erwinia carotovora sub-sp. carotovora. Four isolates of Erwinia carotovora, isolated from cabbage, potato and carrot and one isolate of soft rot causing Rhizopus sp. were evaluated for their ability to produce pectinases. The results showed that carrot isolate of Erwinia carotovora showed the highest activity of pectinases. Rhizopus isolate showed least pectinase activity as compared to bacterial isolates. Studies on host range of pathogen under artificial conditions revealed that all the seven vegetables species tested, viz., knol khol (Brassica oleracea var caulorapa), potato (Solanum tuberosum), onion (Alium cepa), carrot (Dacus carota) and cucumber (Cucumis sativa) were infected by the pathogen with varying intensities. Experiments conducted to standardize optimum temperature for growth of the pathogen revealed its ability to grow at temperatures ranging from 15 to 40 oC. The highest growth of the inoculated broth was recorded at 30oC (optical density 0.273) followed by 25 oC and 35 oC, respectively. Soft rot development also varied with prevalent temperature. Maximum soft rot (39.99 per cent) was recorded at a temperature of 30oC followed by 35oC. In-vitro studies on bioassay of chemicals against Erwinia carotovora sub-sp. carotovora reveal that Ciprofloxacin proved highly inhibitive to bacterial growth by producing highest inhibition zone (21 mm) followed by penicillin, tetracycline and oxytetracycline. These antibiotics were selected for post- harvest management of the disease in storage and the studies revealed that Ciprofloxacin at 200µg ml-1 was most effective when used as pre, simultaneous, and post inoculation sprays of cabbage heads. Other three antibiotics viz., penicillin, tetracycline and oxytetracycline also gave good control of the disease. Cabbage heads treated with ciprofloxacin (200µg ml-1) packed in net bags showed lowest intensity of 4.5 % with a weight loss of 3.01 % due to maceration after six days of storage, where as cabbage heads packed in poly bags without any chemical treatment showed highest rot intensity of 77.78 percent with a weight loss of 69.50 per cent due to maceration during the same course of time