This paper mainly focuses on the optimized methods of the sprinkling irrigation for greenery patches, by maximally equalizing the amount of water sprayed on a certain area. Various models are being discussed, where the main mathematical tool is analytic geometry, employed to research the possible effects of different proposals.
Firstly, the simplest models are built based on a totally ideal situation. Assuming that sprinkling spouts are spinning over plain lawn with a set of specified radii, install them in arrangements of simple geometric figures. Areas of overlapping and blank parts are being calculated and the most reasonable arrangement of all that are studied is selected.
Secondly, real factors are taken into consideration separately as follows: 1. The disequilibrium of the water that drops in a line from the sprinkling center is transformed into a functional expression, whose graphs are drawn to show the water distributed over the area; 2. The plane models are changed into solid ones on the assumption that the sprinkling spouts are placed on slopes. Analytic geometry methods are employed to describe the range of sprayed water on the oblique surface. Through calculation and analysis, models can be adjusted to specific situations.
Finally, the boundary problems and landscape effects are involved.
This paper researches on the judgment theorem and proof of the equivalency condition of a class of symmetric inequalities. By controlling two elementary symmetric polynomials and using the monotonicity of functions and Jensen inequality, it finds the necessary and sufficient condition of the equivalency a class of three-variable and n-variables symmetric inequalities. And we illustrate the application of this method in proof of these inequalities. Then we obtain several judgment theorems on symmetric and cyclic inequalities.
This article studied the speed of addition and multiplication of natural number in different scale systems, the times of addition and multiplication in different scale systems have been compared quantificationally through a function model presented. The conclusion is that binary system is the best for addition, binary system and ternary system are better than other scale systems for multiplication, and they each has a suitable range.
Heavy Snow will turn to a natural disaster, and will bring big economic losses. The authors hope to establish a mathematic model and a plan to illustrate how to sweep off the snow on main roads in a city so as to ensure the smooth flow of traffic. The research will become a basis on which the government can workout some plans against the snow disasters.
Firstly we made some assumptions, then studied the working process of snow-sweepers and derived a snow sweeping model. Based on this model, the relationship between working speed of snow-sweeper and snow thickness, a working model of snow-sweeper, and minimum sets of snow-sweeper needed were established. A deep-searching model and a model of snow-sweeping tree, and a computer program in PASCAL to determine the minimum number of snow-sweeper and the snow sweeping plans on main roads were obtained.
We tried several solutions to deal with this problem, such as using piecewise curve fitting to transform a formula h~ vc relating to a cubic equation operation to formula vc~ h ,using trial and error method to determine the minimum number of snow sweepers, using deep searching and tree structure in one-stroke processing in a complicated two-way road system, and also a matrix to deal with the deep searching.
An order 3 magic hexagon resembles the shape of a 19-cell honeycomb, arranged in a 3 4 5 4 3 manner. The requirement is to fill the numbers 1-19 in the grids so that each row (15 in total) adds up to 38.
Previously invented methods aimed at solving this problem and proving its uniqueness were either not rigorous enough or too intricate. So by analyzing its properties, I wanted to find a combinatorial solution to its construction, prove its uniqueness, and investigate whether its mathematical principles can be used in real-world applications. The difficulty depends on the viewpoint, so the first step was to label each grid in a convenient way. I chose to look at the magic hexagon as a network composed of a center and rings. Then the connections and restrictions of each number set could be found by formula
derivation. In a similar fashion, symmetrical properties were also found. The next step was to analyze possible distributions of odd and even numbers. Out of the 9 configurations, only 1 proved to be usable. The final step was construction. With all the properties known, the few impossibilities were easily eliminated, and only one solution remained, thus proving its uniqueness.
The procedures used on the order 3 magic hexagon may be extended to those of higher orders, providing more ease in their construction. The unique properties of magic hexagons may be used in some fields of application, such as in password systems, large-scale roof structure, composite material, national security systems and many other fields.