专业英语翻译好的材料.pdf

1、Introduction Lights,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“0s+1s=rrQxQxQx2211”is more complicated.It s wise to use the program or mathematical Software such as Mathematic.2、The light issue when n=9 For 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 nngrid checkerboard s initial state vector0s,the terminated state vector 1sand the initial control matrix A.,whether we can attribute the feasible method of Judgment which transforms 0sinto1sto the existence of solutions of equations over finite fields 2Fby clicking on the grid:whether there isixwhich beyond 2F(li,.,3,2,1)to make the equation 1102)(saxsniiibecome true.If there is ixwhich beyond 2F,we can transforms 0sinto1sby clicking on the grid.However,If there isn tixwhich beyond2F,we canttransforms 0sinto1sby clicking on the grid.In order to solve the variableixof the equation1102)(saxsniii,We transform it into a matrix equation form.Namely,we solve the variable x of the equation01ssxAin which A is equal to 221naaa.In Figure 1,the initial state vector of the window is represented by0s.Therefore,0sis equal to,)1,0,0,0,0,0,0,0,1,0,1,1,1,1,1,1,1,0,0,1,1,1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1(TThe terminatedstate vector of the window is represented by1s.Therefore,1sis equal to,)1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(TWe can list the equation 0181812211ssaxaxaxaxiiin which1s+0sis equal to,)0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,1,0,0,0,1,1,1,0,1,1,0,1,0,0,0,0,0,1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,1,01,0,0,0,0,0,0,1,0,1,1,1,1,1,1,1,0(T+z+zThe coefficient matrix A is HEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEH1111111111111111111111111H111111111EBy the elementary row transformation,we can obtain the equation 987654321IIIIIIIIIHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEH12345678923456789INNNNNNNNHEOOOOOOOPOEOOOOOOPOOEOOOOOPOOOEOOOOPOOOOEOOOPOOOOOEOOPOOOOOOEOPOOOOOOOEPOOOOOOOO，TI)0,1,1,1,1,1,1,10(1，TI)1,0,0,0,0,0,0,01(,2，,TI)1,0,1,0,0,1,0,01(3，,TI)1,1,1,1,1,1,1,01(4，,TI)1,0,1,0,0,0,0,01(5，TI)1,0,0,0,1,1,1,01(,6，,TI)1,0,0,0,0,1,0,01(7，,TI)1,0,0,0,0,0,0,01(8，,TI)0,1,1,1,1,1,1,10(9，.So the original equation 0181812211ssaxaxaxaxiican be written in the form below)91(,919293949596979899876543299876543219ixxxxxxxxxXIIIIIIIINXXXXXXXXXHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEOOOOOOOEHEPOOOOOOOOiiiiiiiiii，其中Through a direct derivation,we can obtain the equation.2,21nkPPHPkkk.1,110nkIPNikiikTherefore,98322119IPIPIPIN=T)1,0,1,0,1,0,1,0,1(Through the mentioned above,we can obtain the equation 999NXPAnd also can solve the equation:9X=Txxxxxxxx),(80787674817977759XParticular solution is TTxxxxxxxxxX)0,0,0,0,0,0,0,0,1(),(8180797877767574739sSimilarly,we can obtain the equation TX)0,0,0,0,0,0,0,1,1(8TX)0,0,1,1,1,1,0,0,1(7TX)1,0,1,0,0,0,0,1,1(6TX)0,0,0,0,0,0,0,0,0(5TX)0,0,0,0,0,0,0,1,0(4TX)1,1,1,1,1,1,0,1,0(3TX)1,1,0,1,1,1,0,0,0(2TX)0,1,1,1,0,1,1,1,1(1So we can find a particular solution:of the equations:。Tx)0,0,0,0,0,0,0,0,1,0,1,1,1,1,1,1,0,1,0,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,1,1,0,1,10,1,0,0,1,1,1,0,0,0,0,1,1,1,1,0,1,1,1,1(Now we can find all eight linearly independent solutions of the equation 08181xA.the original equations can be turned into the form:12345678998765432123456789INNNNNNNNXXXXXXXXXHEOOOOOOOPOEOOOOOOPOOEOOOOOPOOOEOOOOPOOOOEOOOPOOOOOEOOPOOOOOOEOPOOOOOOOEPOOOOOOOOThrough solving the equation,we can obtain the fundamental System of Solutions:基础解系1X.)0,0,0,0,1,0,1,0,1(;)0,0,0,1,0,1,0,1,0(;)1,0,1,0,0,0,0,0,0(;)0,1,0,1,0,0,0,1,0(;)0,0,1,0,1,0,1,0,1(;)0,1,0,0,0,1,0,1,0(;)1,0,0,0,0,0,1,0,0(;)0,1,0,1,0,1,0,0,0(TTTTTTTT2X.)1,1,0,0,0,0,0,1,1(;)0,0,1,1,0,1,0,1,1(;)1,0,1,1,0,0,0,0,0(;)1,1,0,1,1,0,1,1,1(;)0,1,1,0,1,0,1,0,1(;)1,1,1,0,1,1,0,1,1(;)1,1,0,0,0,1,1,1,1(;)1,1,0,1,0,1,1,0,0(TTTTTTTT3X.)1,0,1,0,0,0,1,0,1(;)0,1,0,1,0,0,0,1,0(;)0,0,1,0,1,0,0,0,0(;)0,1,0,1,0,0,0,0,0(;)0,1,0,1,0,0,0,0,0(;)0,0,0,0,0,1,0,1,0(;)1,0,1,0,1,0,0,0,1(;)0,1,0,0,0,1,0,1,0(TTTTTTTT4X.)0,1,1,1,0,1,1,1,0(;)1,1,1,0,1,1,1,0,0(;)1,1,0,1,1,1,0,0,0(;)0,0,0,0,0,0,1,1,1(;)1,1,0,1,1,0,1,0,1(;)1,1,1,0,0,0,0,0,0(;)0,1,1,0,1,0,1,0,1(;)0,0,1,1,1,0,1,1,1(TTTTTTTT5X.)0,0,0,0,0,0,0,0,0(;)0,0,0,1,0,1,0,0,0(;)0,0,1,0,0,0,1,0,0(;)0,1,0,1,0,1,0,1,0(;)1,0,1,0,0,0,1,0,1(;)0,1,0,1,0,1,0,1,0(;)0,0,1,0,0,0,1,0,0(;)0,0,0,1,0,1,0,0,0(TTTTTTTT6X.)0,1,1,1,0,1,1,1,0(;)1,1,0,1,1,0,0,0,0(;)1,0,1,0,1,0,1,1,0(;)1,1,0,1,0,1,1,0,0(;)0,1,1,0,1,1,0,0,0(;)0,0,1,1,0,1,0,1,1(;)0,0,1,1,0,1,0,1,1(;)0,0,0,0,1,1,0,1,1(TTTTTTTT7X.)1,0,1,0,0,0,1,0,1(;)0,0,0,1,0,0,0,0,0(;)1,0,0,0,1,0,1,0,1(;)0,1,0,0,0,1,0,0,0(;)0,1,0,0,0,0,0,0,1(;)0,0,0,1,0,0,0,1,0(;)0,0,0,0,1,0,1,0,0(;)0,0,0,0,0,1,0,0,0(TTTTTTTT8X.)1,1,0,0,0,0,0,1,1(;)1,1,1,0,0,0,0,1,1(;)0,1,1,1,0,0,0,0,0(;)0,0,1,1,1,0,0,1,1(;)0,0,0,1,1,1,0,0,0(;)0,0,0,0,1,1,1,0,0(;)0,0,0,0,0,1,1,0,1(;)0,0,0,0,0,0,1,1,1(TTTTTTTT9X.)1,0,0,0,0,0,0,0,1(;)0,1,0,0,0,0,0,0,0(;)0,0,1,0,0,0,0,0,1(;)0,0,0,1,0,0,0,0,0(;)0,0,0,0,1,0,0,0,1(;)0,0,0,0,0,1,0,0,0(;)0,0,0,0,0,0,1,0,1(;)0,0,0,0,0,0,0,1,0(TTTTTTTT3、the classification of the board of the ninth order Similar to the board of the fifth order,the relation of jf(1 j 81)and iV(1 i56)is the chart below:kf所属集合k所属集合kf所属集合k所属集合kf所属集合k所属集合1V1U=41 20V20U=63 39V39U=12,28 2V2U=74 21V21U=72 40V40U=40,42 3V3U=75 22V22U=44,66 41V41U=33,65 4V4U=56,76 23V23U=8 42V42U=15 5V5U=5,37,45,77 24V24U=4,20 43V43U=25,73 6V6U=62,78 25V25U=6,26 44V44U=16,36 7V7U=79 26V26U=29 45V45U=52 8V8U=80 27V27U=67 46V46U=53 9V9U=21,81 28V28U=39,43 47V47U=47 10V10U=22 29V29U=27 48V48U=34 11V11U=58 30V30U=9,57 49V49U=32,50 12V12U=3 31V31U=14,68 50V50U=64 13V13U=23,59 32V32U=2 51V51U=11,51 14V14U=19 33V33U=69 52V52U=10 15V15U=38,44 34V34U=55 53V53U=17,49 16V16U=60 35V35U=54,70 54V54U=18 17V17U=7 36V36U=48 55V55U=30 18V18U=1,61 37V37U=31,71 56V56U=35 19V19U=24 38V38U=13 This gives a partition of set V:V=iiV4,3,2,1and jiVVji,.So we can derive the general rules of every specific elementiV)561(k.The chart3 give a case 表 3 2m3m4m5m6m7m8m9m10m11m12m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13m14m15m16m17m18m19m20m21m22m23m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24m25m26m27m28m29m30m31m32m33m34m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35m36m37m38m39m40m41m42m43m44m45m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46m47m48m49m50m51m52m53m54m55m56m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 1mhave no impact on t.we can give the general rules of everyspecific elementiS)561(k.17 31 11 23 04 24 16 22 29 51 50 38 37 30 41 43 52 53 13 23 08 09 12 18 42 24 28 38 25 54 36 48 40 47 55 43 04 14 27 39 39 27 14 04 21 46 35 52 48 50 44 45 34 33 03 29 10 12 15 17 05 19 n0v关We can derive the general rules of every specific elementiS)561(kin which the general rules of iSis in the chart4 below:表 4 1S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、conclusion The model transforms the light issue into the solving problem of equationsover 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.49 40 21 26 30 32 34 36 20 42 01 02 03 04 05 06 07 08