专业英语翻译好的材料.doc

1、IntroductionLights, which are popular in a very interesting game on the Internet in recent years, are defined as follows: In an n n grid board, every square have two states: white (open) and black (closed). When you click on any one of these squares with the mouse, the box and all its adjacent boxes change states. Namely, the black boxes become the white boxes. At the edge of the board, the box cant have the four adjacent boxes. Therefore, we only consider those existed boxes.using the method of algebra and mathematical modeling to give the mathematical modeling, based on the linear equations of finite field, Zhou Hao gave all the solutions for the case of n=5.Furthermore, he used the Classification method of algebra to give the intuitive description of four equivalence classes of the game so that the players can immediately determine which type the “mess” is.Of course, before Zhou Hao, scholars added the control vector for a problem of PRG game on a similar light and analyzed and proved the problem by a mathematical model. After that, researchers made general promotion for a control problem in the RPG game and give a more comprehensive solution.Furthermore, some foreign scholars used the dynamic programming methods to prove a variety of matrix reconstruction problems and also proved some complex results on reconstructing neighborhood binary matrix.The mathematical knowledge that Zhou Hao use to build the model on the control state issues of lights is relatively complicated. After that, scholars made general promotion for the Zhou Haos mathematical model in order to be accepted. However, if the control variable is larger, solving the effective matrix “Q” and the combination “+=” is more complicated. Its wise to use the program or mathematical Software such as Mathematic.2、The light issue when n=9For the above defined rules of the game, we study the following two issues:When n=9, the boards initial state is a mess: part of the boxes is white and part of the boxes is black. If you continue to click on it, whether there is a way to make the mess eventually become completely white or completely black.Such light issues are control issues. Because light only have opened or closed states, we can make the definition such as binary vector to study the issue, and ultimately translate it into the existence of linear equations on a limited domain.Under the conditions of the Known grid checkerboards initial state vector, the terminated state vector and the initial control matrix A., whether we can attribute the feasible method of Judgment which transforms into to the existence of solutions of equations over finite fields by clicking on the grid: whether there iswhich beyond () to make the equation become true. If there is which beyond , we can transforms into by clicking on the grid. However, If there isnt which beyond, we cant transforms into by clicking on the grid. In order to solve the variableof the equation, We transform it into a matrix equation form. Namely, we solve the variable x of the equation in which A is equal to .In Figure 1, the initial state vector of the window is represented by. Therefore, is equal to The terminated state vector of the window is represented by. Therefore, is equal to We can list the equation in which+ is equal to The coefficient matrix A is By the elementary row transformation, we can obtain the equation ，, .So the original equation can be written in the form below Through a direct derivation, we can obtain the equation Therefore, =Through the mentioned above, we can obtain the equation And also can solve the equation: =Particular solution is Similarly, we can obtain the equation So we can find a particular solution: of the equations: Now we can find all eight linearly independent solutions of the equation .the original equations can be turned into the form: Through solving the equation, we can obtain the fundamental System of Solutions:基础解系3、the classification of the board of the ninth orderSimilar to the board of the fifth order, the relation of (1j 81)and (1i56) is the chart below:所属集合k所属集合所属集合k所属集合所属集合k所属集合=41=63=12,28=74=72=40,42=75=44,66=33,65=56,76=8=15=5,37,45,77=4,20=25,73=62,78=6,26=16,36=79=29=52=80=67=53=21,81=39,43=47=22=27=34=58=9,57=32,50=3=14,68=64=23,59=2=11,51=19=69=10=38,44=55=17,49=60=54,70=18=7=48=30=1,61=31,71=35=24=13This gives a partition of set V: V = and . So we can derive the general rules of every specific element. The chart3 give a case表3奇数奇数偶数偶数偶数奇数奇数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数奇数偶数奇数奇数偶数奇数奇数奇数偶数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数奇数奇数偶数奇数偶数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数奇数偶数奇数偶数偶数偶数奇数偶数偶数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数奇数奇数奇数偶数奇数偶数奇数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数Here the value of have no impact on t. we can give the general rules of every specific element. 1731112304241622295150383730414352531323080912184224283825543648404755430414273939271404214635524850444534330329101215170519494021263032343620420102030405060708We can derive the general rules of every specific elementin which the general rules of is in the chart4 below:表4 中元素的规律N(01)N(02)N(03)N(04)N(05)N(06)N(07)N(08)N(09)N(10)N(11)奇数奇数偶数偶数偶数奇数奇数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数N(12)N(13)N(14)N(15)N(16)N(17)N(18)N(19)N(20)N(21)N(22)偶数奇数偶数奇数奇数偶数奇数奇数奇数偶数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数N(23)N(24)N(25)N(26)N(27)N(28)N(29)N(30)N(31)N(32)N(33)偶数偶数奇数奇数偶数奇数偶数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数N(34)N(35)N(36)N(37)N(38)N(39)N(40)N(41)N(42)N(43)N(44)偶数奇数偶数奇数偶数偶数偶数奇数偶数偶数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数N(45)N(46)N(47)N(48)N(49)N(50)N(51)N(52)N(53)N(54)N(55)奇数奇数奇数偶数奇数偶数奇数偶数奇数奇数奇数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数偶数Through the chart4, we must be able to click with the mouse to transform the state of chart1 into completely white. Based on the chart4, we can pick a form of the most simple state as a representative from each of the specific (i = 1, 2, 3, 4).we can called them type I, type II,.,type 56 in which type I is completely white.4、conclusionThe model transforms the light issue into the solving problem of equations over finite fields and we can give a promotion. However, when the control matrix A class is larger, the solution is complicated. We can only use the computer program to solve it.