The Line between Reasonable and Unreasonable Force Essay

The Line between Reasonable and Unreasonable Force - Essay Example The main argument would be that he/ she used more than reasonable force, but when we look at it in terms of punches and kicks, how is one supposed to tell how much force will stun someone, or kill another person. From this, one cannot help but notice how sketch an equivocal the definitions can be. A real-life example of this is also seen in George Zimmerman’s case he was arraigned for murder because he shot to death a youth who was beating him up. According to his lawyer, he will not go for the stand your ground law defense as many expected which states that: â€Å"A person who is not engaged in an unlawful activity, and who is attacked in any other place where he or she has a right to be has no duty to retreat and has the right to stand his or her ground and meet force with force, including deadly force if he or she reasonably believes it is necessary to do so to prevent death or great bodily harm to himself or herself or another or to prevent the commission of a forcible fe lony†. Despite this law being more recent than the retrospective self-defense one, it still falls short in representing my hypothetical killer, as well as Zimmerman. This is because the use of the term reasonable assumes the participants will be in a position or will have time to reason. In most cases, it comes down to the initial victim's perception which is often â€Å"my life or theirs†. It also does not protect one from the bias that often results based on the age, reputation an often the race of the final victim.

Black People and African American Actors Essay Example for Free

Black People and African American Actors Essay â€Å"We black men have a hard enough time in our own struggle for justice, and already have enough enemies as it is, to make the drastic mistake of attacking each other and adding more weight to an already unbearable load. † (Malcolm X) African American men through time have struggled for a power that is out of their reach because others hold the power. August Wilson’s Fences displays a Psychological/Psychoanalytic approach by illuminating the inherent injustice in America’s treatment of African American males and the ways in which this racism affects and invades the societal units – the family. The conventional husband-wife and father-son conflicts are subservient to the plays discussion of racism. Fences is a drama that focuses on the characteristics of black life in a small neighbor in 1957 and the strains of society of the Maxson family. The play shows how the main character struggles against his oppressive past and his present surroundings, and when he tries to regain the power in his life, he fails, and ends up bringing down other with him. The protagonist, Troy Maxson is a restless trash-collector and former baseball player for the Negro League. In the play, Fences, Troy’s past dictates the kind of man he is today. His father, an abusive unsuccessful sharecropper, has had a major impact on Troy. Troy states, â€Å"But I’ll say this for him†¦he felt a responsibility toward us. † (1310; all page references are to class text Literature an Introduction to Reading and Writing, 5th ed. ) This comment is the one good thing that Troy has to say about his father. Despite his father’s meanness, he did feel a sense of duty toward his family. After an incident with his father beating him unconscious, Troy left home at the age of fourteen (Wilson 1311). Leaving the oppressive rule of his father should have brought Troy a feeling of freedom, but Troy found the exact opposite. Troy found that there were no jobs or places for blacks to live so he began stealing to survive. He met a woman and got her pregnant with his first son. Lyons. The responsibility weighed on him because now he had two more mouths to feed (Wilson 1311). To take care of his family, he continued to steal which ended him jail for fifteen years and this is where he learned to play baseball. Troy Maxson was a great baseball player, at least according to his friend Bono. Although he played brilliantly for the â€Å"Negro Leagues†, by the time that blacks were allowed into the Major League Troy was too old. In Troy’s self-created illusion, he believes that he would have made it to the Major League if it were not for the color of his skin (Wilson 1292). Because he never earned the recognition or the money, which he felt he deserved, the discussion of professional sports will often send him into a tirade. In a discussion with Bono and Rose concerning Cory recruitment by a college football team, Troy states, â€Å"Jackie Robinson wasn’t nobody. I’m talking about if you could play ball then they ought to have let you play. Don’t care what color you were. † (1. 1. 78). Troy points out the blatant racism that kept him from a career in the major leagues. He was just as good, if not better, than many of the white players, and yet he did not get a shot. Troy, now fifty-three, has long retired from baseball; he makes a living for himself and his family as a garbage man, and nurses his well-earned bitterness. His life has been warped by white racism, and in turn, Troy is helpless to keep himself from warping his son’s life. Because of the racism, Troy has suffered in life and the tumultuous relationship with his father, Troy tries to control both is son’s lives. Troy has a low expectation of what black men can do with their lives, and is holding his sons back from obtaining successes that Troy could only dream about obtaining. Lyons is ambitious talented jazz musician. Lyons jazz playing appears to Troy as an unconventional and foolish occupation. In the beginning of Fences, Lyons comes to Troy to borrow ten dollars because he girlfriend Bonnie has a job working at the hospital. In Troy’s mind, Lyon is failing in his duty as a man by not taking care of his woman. Troy lectures Lyons, â€Å"I done learned my mistakes and learned to do what’s right it. You still trying to get something for nothing. Life don’t owe you nothing. You owe it to yourself. † (1. 1. 145). The quotation is an example of how Troy feels the black man will never amount to anything in the â€Å"white man’s world†. He also tries to control his son, Cory’s future because he see that he is going down the same road the Troy was on and was rejected from. Troy tells his wife Rose â€Å"The white man ain’t gonna let him get nowhere with the football. † (1. 1. 65). Through racial discrimination is still a huge problem in America during the 50s, things have gotten more equal, especially in the world of sports. Troy however is too stubborn and bitter to admit there has been some progress. Troy is now a fifty-three year old African American male who works for the sanitation department. Troy works to gain power as a man by changing his job situation. Troy goes to his boss, Mr. Rand and asks him â€Å"Why? Why you got white mens driving and the colored lifting? † (1. 1. 10). Troy sees this as oppression, though he knows that he has a job that is awarded to both white and black men, the racial line is till carefully drawn. He is determined to cross this line because he cannot handle any racial prejudices in his life. Troy continues to recount his fight with his boss to his long-time friend, Bono. Troy states, â€Å"You think only white fellows got sense enough to drive a truck. That ain’t no paper job! † (1. 1. 10). Troy feels that blacks are good enough to be drivers; he assumes that they would not be able to handle â€Å"paper† or office jobs. This is another example of how racism is so entrenched that black people are a little racist against themselves. Troy’s characters sought after this job in order to gain back some of the power in his life, even though he does not have a license to drive. He is finally able to prove to himself and the people around him that he is worthy and just as good as any other man, including white men. As a father, Troy feels obligated to provide the necessities of life, but he seems to think his duties end there. During a discussion between Cory and Troy, when Cory asked he father, â€Å"How come you never liked me? † Troy replies, â€Å"You live in my house†¦sleep your behind on my bedclothes†¦fill you belly up with my food†¦cause you my son. You my flesh and blood. Not cause I like you? Cause it’s my duty to take care you. (1. 3. 107). It is clear the legacy of racism threatens to take another generation. However, he was unable to provide for his family alone and had to use the tragic injuries of his brother, Gabriel, a World War II veteran. Troy uses this money to pay for his house. He says, â€Å"If my brother didn’t have that metal plate in his head†¦I wouldn’t have a pot to piss in or a window to throw it out of. And I’m fifty-three years old. † (1. 2. 64). He feels weighed down by the guilt of using his brother’s destruction as a way to jumpstart his own life. He feels like the only reason he has anything is the fact that his brother life was ruined. Through he has a loving relationship with his wife Rose, he still strays and finds a woman with which he feels he can be a different man. He uses this affair as a way to escape from the responsibility of his life and the constant reminders of his shortcomings as a man. When trying to explain why he had the affair he tells Rose, â€Å"I can step out of this house and get away from the pressures and problems†¦be a different man. I ain’t got to wonder how I’m gonna pay the bills or get the roof fixed. I can just be a part of myself that I ain’t never been. † (2. 1. 98). His affairs results in a baby, and he needs Rose to help him, after his mistress, Alberta, died. Rose accepts the responsibility of raising this child, but she has cut off all emotion bonds with Troy. Rose tells him â€Å"I’ll take care of you baby for you†¦Ã¢â‚¬â„¢cause like you say†¦she’s innocent†¦and you can’t visit the sins of the father upon the child. From right now†¦this child got a mother. But you a womanless man. † (2. 3. 4). It is with this act that Troy loses more of his control and finds himself even more emasculated than before. Troy was unable to see that he did have control over one aspect of his life, yet he was too worried about proving his worth to others and to himself, that he proved his unworthiness to his own wife. Troy is a resentful victim of his life’s circumstances, a man who has become fenced in from happiness by the conviction that he was never paid what he was owed in any right: not from his father, not from his former baseball career, not from his employers, and not from his family. Troy tried to take back the power that was taken from him by demanding that his loved ones live practical, responsible lives while he has the freedom to have an affair, rebel against racist practices of his employers by protesting the limitation of black workers as lifters not drivers on the trash trucks. Troy refuses to see life in any presented to him but the way he perceives events in his own head. Eventually, Troy’s death leaves many negative attributes an inheritance for his family to sort out and accept. Fences I. INTRODUCTION (4-6 sentences) A. Quote B. Thesis Sentence–The play illuminate the inherent injustice in the America’s treatment of black men and the ways in which this racism affects and invades the family. 1. The conflicts in the play show has racism affected the family. 2. Set in 1957 after World War II when blacks still are fighting for power. 3. Shows how the characters struggle against his oppressive past and present, and when he tries to regain power he fails and brings down others around him. II. BODY (minimum of three paragraphs) A. 1st Supporting Paragraph (8-12 sentences) 1. Topic Sentence–Troy’s past dictates the kind of man he is today a. Subject The ways he was raised b. Focus – why he left home and what he did when he left home 2. 1st Subtopic Sentence–His father was an abusive unsuccessful sharecropper 2nd Subtopic Sentence–left home at an early age and landed in jail. 3. Concluding Sentence– uneducated black man did what he had to do to serve. B. 2nd Supporting Paragraph (8-12 sentences) 1. Topic Sentence–Troy’s injustice dealing with his career a. Subject–Lost opportunity due to racism towards blacks in sports b. Focus– why is lost his opportunity to be the man he thought he was supposed to be and how he handles it 2. 1st Subtopic Sentence– during that time blacks were only allowed to play in the Negro League. 2nd Subtopic Sentence–Any talk about sports through Troy into a tirade. 3. Concluding Sentence–Because of the oppression that was placed on him, it causes he to oppress others dreams C. 3rd Supporting Paragraph (8-12 sentences) 1. Topic Sentence–The affect his past has on him and sons a. Subject–Troy’s low expectation for options that the black man has in a career b. Focus–He holds his sons back from realizing their dreams 2. 1st Subtopic Sentence–Thinks he son Lyon is wasting his life by not accepting a job that is thought as acceptable in the white man world 2nd Subtopic Sentence–Does not want his son Cory to follow in footsteps as an athlete because of how he was treated. 3. Concluding Sentence–Troy is too stubborn and bitter to admit there have been changes since his time concerning treatment of black athletes D. 4th Supporting Paragraph (8-12 sentences) 1. Topic Sentence–His job and racism a. Subject–Whites has the better job in the sanitation department b. Focus–Tries to change his job situation 2. 1st Subtopic Sentence–Troy goes to his boss and confronts him about the racism at work 2nd Subtopic Sentence–Troy himself is racist toward other blacks 3. Concluding Sentence–Troy is able to prove to himself and family that is worthy and as a good as any white man. E. 5th Supporting Paragraph (8-12 sentences) 1. Topic Sentence–Responsibilities as a father a. Subject–Troy believes that a father is only supposed to provide for his family b. Focus–how he goes about it 2. 1st Subtopic Sentence–Tells he son that he doesn’t have to love him he just has to make sure that he has a roof over his head 2nd Subtopic Sentence–Provides for his family by using his brother 3. Concluding Sentence– F. 5th Supporting Paragraph (8-12 sentences) 1. Topic Sentence–The effect it has on him relationship with his wifea. Subject–has an affair b. Focus–blames his responsibilities on his affair 2. 1st Subtopic Sentence–Tells his wife that he now choice but to cheat because of the things he has to do to make a life for his family 2nd Subtopic Sentence–Loses he wife 3. Concluding Sentence–is further emasculated III. CONCLUSION (3-4 sentences) A. Troy is a victim of life: his father, his dead career, his job and family all play are part B. Troy tries to control his life but the decision he makes lead him to be more oppressed then he was before. He is stuck in the past and cannot move forward with his life. His death continues the cycle.

The Mystery Behind Sudoku Mathematics Essay

The Mystery Behind Sudoku Mathematics Essay Puzzle games can be very enjoyable and is popular amongst kids as well as adults. Many of you may know the game Sudoku; where by the goal of the game is to fill in the remaining empty cells with each number from 1-9 appearing no more than once from each column, each row and each of the nine sub-grids. Sudoku is a type of logic-based numerical puzzle game that has a unique solution once completed. The most common form of a Sudoku is constructed as a 99 grid with nine 33 sub-grids and is primarily partially completed. Sudoku has become appealing among puzzle enthusiasts and involves complex thinking and practice. Available daily in newspapers, mobiles and many more, this addictive and brain-teasing puzzle game has become one of the most popular games to play since the time of the Rubiks cube. This dissertation discusses the mathematical side involved in Sudoku. There is no mathematics in actually solving a Sudoku but more of how it is used from a creators side. The 99 grid will be considered in the majority of the report; however a glimpse into other size grids will be discussed briefly also known as variants. Mathematicians have been questioning How many unique solutions are there in a Sudoku? Essentially meaning what are the possible ways of filling in an empty Sudoku grid so that each row, column and sub-grid contains the numbers 1 through 9. Your first thought of an answer may be a couple of thousands, but as you understand the concepts behind a Sudoku, you begin to grasp a whole new aspect. Combinatorics and permutation group theory are largely interwoven with analysing Sudoku. For that reason, I aim to explore these theories and understand how it applies to the methods of enumerating Sudoku grids. In particular I will be looking at Felgenhauer and Jarviss approach to enumerating all possible Sudoku grids where they employ several mathematical concepts. Furthermore I will uncover the importance of Latin squares and its use of constructing Sudokus. There are many constraints in regards to when are similar solutions considered different such as solutions of similar structure, symmetry etc. Preserving symmetries are known as relabeling symbols, band permutations, reflection, transposition and rotation. Burnsides Lemma theorem is one of their techniques in computing the number of essentially different solutions. Many difficult problems are of the type called nondeterministic-polynomial known as an NP-complete problem. This will direct me onto the debate on whether Sudoku is an NP-complete problem. Sudokus can take many forms and shapes. These are called Sudoku variants and consist of rectangular regions, Sudokus with a large region having no clues (numbers), an empty row, column or sub-grid and many more! Here I will research the logic behind irregular Sudokus as well as examining any occurring patterns or whether it has occurred by chance. 1.2 Latin squares and Sudoku Sudoku is also a special case of Latin squares. The Swiss mathematician, Leonhard Euler made many fundamental discoveries during 1782 including Latin squares. A Latin square is an N x N matrix where by a set of N characters are arranged such that each row and column contains one of each character. This is also in the case of a Sudoku, when complete, with an additional constraint that the nine sub-grids must hold the numbers 1-9. A reduction can be made to any Latin square by permuting the rows and columns. This arrangement is an aspect of combinatorics and is most commonly referred to as enumeration. Enumerative combinatorics is a classic area of Combinatorics and involves counting the number of infinite class of finite sets. Counting combinations and counting permutations are two of the most common forms. The number of valid Latin squares is known to be approximately 5.525 x 10 ²Ãƒ ¢Ã‚ Ã‚ ·. Write about Colbourns proof 1.3 Combinatorics and Permutation group theory Combinations and permutations have slightly different meaning. Combinations are the number of different ways of selecting n objects from a set but the order of events is not important. From a set of 3 objects, lets call these 1, 2 and 3. If for example I was asked to pick the number of ways of selecting 2 objects out of the 3, there would be three combinations 12, 23 and 13. 12 = 21 since the order of each pair is not important. A permutation on the other hand does consider the position. Therefore if I was to use the above example, there would be six permutations. A simpler way to calculate a larger set would be to use formula 1: Formula 1. = = Where is the combination formula, is the permutation formula, n is the total number of objects and r is the number to be arranged Both methods are one way of computing the number of possible Sudoku solutions and this will be looked at later in the report. Chapter 2 Enumerating possible Sudoku solutions 2.1 Distinct Sudoku solutions There are many approaches to enumerating possible Sudoku solutions. To enumerate every possible Sudoku solution, a Sudoku differs from another if they are not identical. Thus all solutions will be consider unless they are like for like. Felgenhauer and Jarvis was the first to enumerate the Sudoku grid solutions directly in 2005. There approach was to analyze the permutations of the top row used in valid solutions. Their knowledge of the complexity in computing the number of Latin squares has made them aware of how they should go about getting an answer with fewer computations. Hence by using relabeling this could shorten the number of counts. To make it easier, each sub-grid is given an abbreviation seen in figure 3. B1 B2 B3 B4 B5 B6 B7 B8 B9 Figure 1. Abbreviated sub-grid with top band (Felgenhauer and Jarvis, 2006) Firstly they consider every solution to filling in blocks B2, B3, given that B1 is in standard form. To work out every possible way of arranging B1 on its own would essentially be computing the number of permutations of 9 symbols. There are 9! of filling in B1. The main operation they use is called relabeling. 1 2 3 4 5 6 7 8 9 Figure 2. B1 in standard form (Felgenhauer and Jarvis, 2006) Felgenhauer and Jarvis have found that B2 and B3 is the same as the transpose of B2 and B3. Therefore the number of ways of arranging B1, B2 and B3 and B1, B2 and B3 to a complete grid is equally the same. This means that computing one set of possibilities will cut down the number of solutions. Inevitably, there are few pairs of B2 and B3 that needs to be worked out and as well as using reduction the number of possibilities for the top band of a Sudoku grid is 9! x 2612736 = 948109639680. The next section involves brute force computation. As running through all 2612736 possibilities would be exceedingly tedious for B2 and B3, Felgenhauer and Jarvis attempts to identify configurations of the numbers in these blocks which give the same number of ways of completing to a full grid. This in return, will cut down the number possibilities. Permuting B2 and B3 in every way such that the result gives a unique solution will preserve the number of complete grids. This is the same for B5 and B6, and B8 and B9. However this changes B1 from its standard form, so an additional relabeling of B1 needs to be performed. Another approach to reducing the number of possibilities is to permute the columns in each block and permute the rows of any block. Reducing the number of possible ways by permuting. Lexicographical reduction Permutation reduction Column reduction As a result of these methods, Felgenhauer and Jarvis have found that there are approximately 6670903752021072936960 à ¢Ã¢â‚¬ °Ã‹â€  6.671 x 10 ²Ã‚ ¹ Sudoku solutions. In light of this result, there are fewer solutions than Latin squares due to the fact that there is that extra restriction of 9 sub-grids. That being said, there will be no shortage of Sudoku puzzles any time soon. Verification of this result has been confirmed by several other mathematicians Ed Russell to be more precise. 2.2 Essentially different Sudoku grids Whether symmetrical Sudoku grids are considered as two separate solutions is another method of enumerating the possible solutions. In this case, the only solutions are ones that are essentially different. Lets say two Sudoku grids are equivalent if one is a transformation of the other by applying any number of symmetries. If however, no such chain of symmetries can occur between two grids, it is essentially different. Two Sudoku grids are the same if we can get from one to the other by applying some sort of symmetry. For instance, take figure 3 4 below; the set of 3s in the first grid can be interchanged by the placements of the set of 1s, effectively producing the second grid. Figure 3. Valid Sudoku grid Figure 4. Another valid Sudoku grid from Figure 1 As well as this, a solution is said to be the same as another if any two columns or rows are swapped. The first column and second column in figure 3 can be exchanged to give figure 5. The two solutions are said to be symmetrical because the transformation still produces a valid Sudoku grid. Figure 5. First and second column swapped from Figure 1. Another form of symmetries includes rotational grids. A rotation of Figure 3 by 90 degrees generates a new valid Sudoku grid shown in Figure 6. Figure 6. Rotational of 90 degrees from figure 1 Any of these operations performed on a valid grid maintains the property of it being valid and this is known as symmetries of a grid. When an object is subject to these operations, certain properties are preserved. An example would be if one performs symmetry on to a Sudoku grid and repeats this operation once more, the final transformation is itself symmetric. In addition a symmetrical object can be transformed back to its original state by another form of symmetry. Performing several symmetries on a Sudoku grid can also be achieved by grouping its neighbouring pair. So the first symmetry can be paired with the second or the second can be paired with the third and so on. The resulting transformation is nevertheless the same either way. From these properties, it is inevitable to say that the set of symmetries of any Sudoku grid form a group. A group is a set G if it satisfies the following properties: CLOSURE If f and g are elements of G, then f ·g is also an element of G. ASSOCIATIVITY If f, g, and h are elements of G, then f ·(g ·h)=(f ·g) ·h must satisfy. IDENTITY ELEMENT There is an element e in G such that g ·e=e ·g=g for all g in G. INVERSE For any element g of G, there is another element d of G such that g ·d=d ·g=e, where e is the identity element. (The element d = g-1.) The symmetry group is thus generated by the transformations of: re-labelling the nine digits, permuting the three stacks (3 vertical blocks of a Sudoku), permuting the three bands (3 horizontal blocks of a Sudoku), permuting the three columns within a stack, permuting the three rows within a band, and any reflection or rotation. These can be combined to form other elements of the group and together they comprise of the symmetry group G. Given that any element of G can be mapped so that it takes one grid to another, we can say that the set of valid Sudoku grids has a finite number of elements. Thus G has finitely many symmetries. The association between symmetrical Sudoku grids are in fact an equivalence relation and satisfies the following three properties: for grids A, B and C in set G Reflexivity A = A Symmetry If A = B then B = A Transitivity If A = B and B = C then A = C Let A be any valid Sudoku grid, we must consider all the grids that are equivalent to a valid Sudoku grid A. To do this, we firstly have to group together grids that are essentially the same so that we can partition the set of grids. This will break the set of Sudoku grids into subsets, with groups that contain no relating elements within each other. The term subset can be called equivalence classes and is denoted by X/G. In any equivalence class, there are elements that are equivalent to each other by symmetry. The total number of elements in X/G is equal to the number of essential Sudoku grids. To calculate the number of essentially different Sudoku grids, we shall look at all the symmetries neglecting the re-labelling of the nine digits for the time being. The number of distinct symmetries founded by Russell and Jarvis (2006) is said to contain 3359232 (pg 4). In this finite group H, we need to take the average number of grids fixed by an element of H, up to re-labelling. Next we need to verify the number of fixed points of all elements in H. Russell and Jarvis have found that there are 275 classes of symmetries using a software package called GAP. It is interesting to note that some of the elements in H have the same number of fixed grids. In other words, we only need to count the number of fixed points for one symmetry for each of the 275 classes. However there exist symmetries in H that have no fixed points. Subsequently, there is no need to count the number of fixed grids for those that have no fixed points. That being said, there are only 27 out of 275 classes that co ntain fixed points, meaning fewer computations. Rotman. J. J (1995) demonstrate that if X is a finite G-set and |X/G| is the number of G-orbits of X, then Formula 2 holds where, for gцG, X is the number of xцX fixed by g (pg 58-61). Using this notion, we have established that the number of valid Sudoku grids is of a finite set and X/G is the number of essentially different Sudoku grids, so we can obtain the number of essentially different Sudoku grids by using the Burnside Lemma Theorem. Formula 2. Burnside Lemma Theorem (Rotman, 1995) Burnside Lemma Theorem is a useful tool when dealing with symmetry with a set of countable objects. When used to enumerate the essentially different Sudoku grid, the set of equivalent grids form an orbit of the symmetric group. The number orbits are essentially the number of different grid solutions. This may sound slightly (ALOT) trickier to compute, nonetheless Russell and Jarvis have shown that the number of essentially different Sudoku grids is 5,472,730,538 with the implementation of Burnsides Lemma Theorem. Chapter 3 Nondeterministic polynomials 3.1 NP-complete and Sudoku Sudokus may relate to a variety of problems, in particularly, whether Sudoku is an NP-complete problem. It is known that NP-complete problems are one of the most complicated cases in NP, also referred to as nondeterministic-polynomial. Its rival, P problems relates to NP as both being in the same complexity class. Mathematicians have yet to solve whether NP-complete problems can be solved in polynomial time or more commonly whether P = NP. Consequently being one of the greatest unsolved mathematical problems. The majority of computer scientists believe that P à ¢Ã¢â‚¬ °Ã‚   NP, as a result would mean that NP-complete problems are significantly trickier to compute than to verify. Unfortunately, nobody has yet found an efficient algorithm, not even with the use of computers available today. A problem is said to be NP-complete when its solution can be proved in polynomial time. And if that problem can be solved in polynomial time, all problems in NP can be solved too. An interesting characteristic of NP-complete problems is that the time frame to solve the problem increases rapidly as the size of the problem gets larger. If that is the case and Sudokus are NP-complete, solving a Sudoku of higher order (say 17 ² x 17 ²) will become increasingly challenging algorithmically then the standard 3 ² x 3 ² version were talking trillions of years. It has been shown that Sudoku does belong to the category of NPC problems by Takayuki Yato of the Univeristy of Tokyo (2003). An exchange for the notation ASP-completeness (shorthand for Another solution problem), led the proof of NP-completeness of ASP. Their proof uses reduction in order to obtain the required polynomial-time ASP from the problem of Latin squares by Colbourn (1984) who has verified, the NP-completeness of ASP of Latin square completion Another accountable source by Provan states that, It is known that solving general-sized Sudoku puzzles is NP-hard, even for square grids with blocks consisting of the sets of rows and columns (Latin Squares) or for p2 x p2 grids with blocks consisting of rows, columns, and the p2 partitioned p x p subsquares. Mathematical programmes such as the 0-1 linear programming and the knapsack problems are also cases of NP-complete problems. A full list of other problems that are NP-complete can be found in Garey and Johnson (1979). Chapter 4 Sudoku Variants 4.1 Variation The classic form of a 99 Sudoku are polyominoes. There are other variations of Sudokus that can be applied to the rules of Sudoku. There are puzzles of the size 66 with 23 regions or a 1212 grid of 43 regions. More so, there are other fascinating Sudoku variants such as Greater than Sudoku. Chapter 5 Personal Critical Review The progress I have made during the duration of this project, have been fairly slow but surely getting there. Having said this on many occasions, I have still not conquered my time management skills! The project started very slow which meant I was behind schedule. Nevertheless my organisational skills have kept me on balance. The GANT chart has been of great help in doing so. What has kept me going throughout this project in particular would be self discipline and motivation. This project has proven that I am capable of working to my own initiative, but also well within a group; my time during the group project. Furthermore, my time on this project has definitely promoted a better mentality of my future ambitions. I have learnt that it is crucial to read a lot, as well as reading as broadly as I can. This in turn have aided in the running of my project. With other coursework deadlines, I made that a priority and had no time to meet with my supervisor. I understand that meeting with my supervisor is equally important because a supervisor is there to encourage and to advice on any difficult obstacles I may encounter. An area of interest to proof whether NP-complete problems can be solved in polynomial time, was left open as future work. This could be the next step of extending this report that little bit further. Chapter 6 Conclusion A challenging problem for further research is to proof whether NP-complete problems can be solved in polynomial time. This has yet to be solved and anyone who has a formal proof will be rewarded $1 million dollars by The Clay Mathematics Institute.

George Orwells 1984 and Today Essay -- Television 1984 Freedom Essays

George Orwell's 1984 and Today TV rots the senses in the head! It kills the imagination dead! It clogs and clutters up the mind! It makes a child so dull and blind. He can no longer understand a fantasy, A fairyland! His brain becomes as soft as cheese! His powers of thinking rust and freeze! An excerpt from Charlie and the Chocolate Factory, By Roald Dahl, 1964 When George Orwell’s epic novel 1984 was published in 1949 it opened the public’s imagination to a future world where privacy and freedom had no meaning. The year 1984 has come and gone and we generally believe ourselves to still live in â€Å"The Land of the Free;† however, as we now move into the 21st Century changes brought about by recent advances in technology have changed the way we live forever. Although these new developments have seamed to make everyday life more enjoyable, we must be cautious of the dangers that lie behind them for it is very possible that we are in fact living in a world more similar to that of 1984 than we would like to imagine. In 1949 when Orwell’s novel was published, television was a relatively new invention. Fewer than 10% of the United States households had a television set in them and at this time programming was limited to mainly news-oriented shows. Many people believed that television would never surpass radio as the chief means of mass communication; they could not have been more incorrect. Presently 98% of the households in the United States have one or more televisions in them. What once was regarded as a luxury item has become a staple appliance of the American household. Gone are the days of the three channel black and white programming of the early years; that has been replaced by digital flat screen televisions connected to satellite programming capable of receiving thousands of channels from around the world. Although televisions and television programming today differ from those of the telescreens in Orwell’s 1984, we are beginning to realize that the effects of television viewing may be the same as those of the telescreens. The telescreens in 1984 served two purposes, surveillance and mind control. Unlike the televisions of our present day, the telescreens in 1984 also served as a device constantly monitoring the citizen’s actions by means of an integrated camera and microphone in addition to broadcasting continuous p... ...her say to us â€Å"No, I’m sorry I can’t do that right now, I’m watching my show.† Americans have ceased to live their own lives and have practically become slaves to their televisions and the corporations that stand behind them. Unlike the citizens of Oceania, we are able to make our own decisions. We can turn off our televisions; we can live our own lives and make our own experiences. We can learn about and do practically anything we want. Most of us do not take advantage of this freedom. In fifty years when my generation has become grandparents, what stories will we have to tell our grandkids? Will they really want to hear about that episode of Friends that we loved so much? Will we really have any knowledge or experiences worthwhile to tell them? Perhaps it won’t even matter. Perhaps our grandkids will be too interested in what they are watching on television to even want to listen to us. Yes we live in the â€Å"Land of the Free,† but until we really start taking advantage our freedom to the fullest and pull ourselves away from the television we are no better off than the citizens of Oceania and the telescreens that surround them as they toil on in their non-eventful lives.

Human Linguistics and the Importance of Second Language Learning in Modern Living Essay

Language is among the most basic necessities for a person to live. Be it thru oral or written communication, it is inherent in every person to connect with the world. As the world becomes more complicated with all the things that are written about speech, language, and communication in general, it reveals itself as an indispensible part of human life. Now, it implicates more needs than the usual. The basic need for understanding and being understood has branched out to issues about global interconnectedness, political and economic dealings, and cultural relations among others. And it continues to take on several roles in nation-building and for the promotion of social mobility and employment. The advent of technology only makes it even harder for people to stop communicating with the rest of the populace. There is a wide variety of languages in the world right now. â€Å"Biologists studying communication among living beings in general [may think that]—all are relatively minor variations on the single theme of Human language† (Anderson, 2005). But from the point of view of a linguistics expert, the variations in types of sentence construction, pronunciation, and other rules governing languages are very important differences that imply causes yet undiscovered. These examples delineate the function of the biologists’ view of communication and the linguists’ of the same process. It is true indeed that there are more than one ways to look at something but seeing the communication process through different lenses provide more a single way to understand the phenomena. In doing so, people are able to form a holistic view of the process of communication and how it affects human life. The Role of Human Language At this point, language takes on bigger roles which may affect greater humanity. From simply being a medium of sharing how people think of or feel as of the moment, it has become rather an important part of creating human identity. And with the individual identities created by language, it becomes the precursor in creating a national identity (through the shared identities of many individuals living in the same geographical location). People might begin to wonder: why human identity? These may be two separate ideas but in fact they are closely related to each other. Language is directly related to the person and his country of origin. We were given the basic knowledge that those who speak Chinese came from China, people who speak Russian are from Russia, and so on. When our ancestors devised names for the different languages around the world, the name of the country speaking that language was used to identify the language spoken. It might not have any implication during those times and was only done to make things easier for people to remember. But now that people have taken the time to analyze the effect of this naming convention, language has indeed become instrumental to creating identity for people. The prevailing notion of people â€Å"’having’ a language, which is equated to ‘belonging’ to an origin group† (Urciuoli, 1995) is perfectly logical. But then again, because of the proliferation of foreign language studies, specific languages cannot be exclusive to just one place or country any more. In particular, English is not only for the people in England or America. English varieties have come about which decentralized the use of the language. While language origins can still be identified, other nationalities already have the capacity to speak the same. This concept will be discussed further in the latter pages. Human Beings in Terms of a Speaker of a Language Unique to Them Being able to speak a language means mastering the abstractions of the language codes. Semantics, as the study of signs, is an important tradition of thought in communication theory† (Stephen W, 2008). Often times, the theory of semantics are used to explain why words are not really the images that we think of but are rather representation of truth. â€Å"The meaning is arbitrarily assigned† (What is Language, 2006). Therefore, the speaker of a particular language is a person who was able to understand and make sense of the abstraction. This abstraction contributes to the creation of the person’s perception and language skills. And if you will notice, the primary language that a person learned is vital in the subsequent languages that he will study. For example, native Korean speakers might be able to study and speak the English language but often times they are not able to eliminate the â€Å"e† sound after every English word they say. This only proves that the prior language learned by a non-native English speaker is able to influence subsequent languages studied throughout his lifetime. Notwithstanding the presence of Language Acquisition Devices, a person may or may not be successful in learning a new language but if he is then the prior influence will be evident in the newly learned language. The Role of Second/Foreign Language Foreign language becomes even more important now that the world is geared towards globalization. With much concerted effort to adopt a common language, which is English as of the moment, all other countries strive to educate their citizens in communicating using that foreign language. â€Å"Marshall McLuhan was the first person to popularize the concept of a global village and to consider its social effects† (Stewart, 2010). Those who fail to appreciate this effort do not acknowledge the fact that in the global village, the barrier-less communication paves the way for three phenomena: 1. Global information dissemination for improved understanding between nations. 2. It is also the appreciation of cultures other than your own. 3. Aids in the self-actualization of the second language learner. The first phenomenon is plain and simple. If you are going to look at the job of the ambassadors, it looks as if it’s all prestige and fun. In fact, it is predominantly driven by hard work. It is extremely difficult to make great relationships with other countries because there will always be a reason to be skeptical of something you are not familiar with. That is the reason why ambassadors find time to learn the language of the countries where they will be assigned: because insisting on speaking a different language might take its toll upon the relations of the ambassador’s country to the country of assignment. The second phenomena are also something simple which has profound implications at the end of the day. Reasons for Speaking Foreign Language Using the logic that language is culture, learning another language is therefore a way to expose one’s self to another group of people’s way of living. As you immerse yourself in the language, the culture goes with it and in a matter of months the foreign language learner will know whether he likes and would want to continue studying that foreign language. The satisfaction As of today, learning a foreign language has become a necessity. It doesn’t have to be a â€Å"dominant† language for it to be learned by anyone. Although English has become a very popular language to study (considering the political and economic influence of the United States to the rest of the world), studying the language of a small country with not much of an influence to the world can happen because of various reasons. For one, travelling often to a particular place would make a person want to study the language of the person. Or probably, a person who plans to study or work abroad needs to learn the language of the country first before he is cleared at the immigration office. And still, there are underlying reasons behind this. It is probably caused by the need to communicate in the native tongue of the people in that country. Communicating through their native language lends them the feeling of being part of the place. Whereas, not knowing the language alienates the person—it becomes obvious that he is an â€Å"outsider† because he doesn’t know how to speak the language. Another reason for studying a foreign language is to increase the cultural capital of a person. This concept was proposed by Pierre Bourdieu, a French sociologist. He argued that â€Å"these ‘advantages’ stem from the institutionalization of ‘criteria of evaluation’ in schools†¦ which are favorable to a particular class or classes† (Bourdieu, 1977). It must be noted that a couple of decades back, learning a second language is exclusive for those in the upper class of the social triangle—being able to speak another language becomes a status symbol. But that was true in the past years. Nowadays, there are many ways for a person to learn another language without having to spend a lot of money. One way is by subscribing to free online language tutorials. Nevertheless, even if this language skill is already available to most people, the concept of cultural capital is still there. And the person who knows more languages has the advantage. A rather intrinsic motivation for learning a new language is the appreciation of that particular language and the culture that is associated with it. Adopting this kind of perspective will make the process of learning a language enjoyable and â€Å"less results-oriented† (Tender, 2008). The motivational approach of this kind of learning allows the second language learner to appreciate the value of the language prior to learning it. It is rather encouraging for the person to learn the language continuously because the benefit is achieved intrinsically. The Role of the Second Language Teacher Apparently, there are â€Å"benefits and advantages to those who take the time to explore the language of a culture different than their own† (What is the Importance of Learning a Second Language? ). And whatever these benefits and advantages are, it is the duty of the second language teacher to make his/her students realize it. The role of a second language teacher takes on a support role which motivates the student every step of the way. By doing so, the students therefore realize the importance of what they are doing and will adopt a learning mechanism in which the rewards are rather intrinsic. It is also not enough for the second language teacher to stop at teaching grammar and syntax rules. More important than teaching the language is teaching them why they wanted to study the language in the first place. Often times, those teachers who elicit a compelling need for their students to study the language are the ones who are most successful with their teaching career. They do not stop at the superficial level of teaching a language but rather delve on the subjective inferences of asking the question â€Å"why†. Also, like any other teacher, a second language teacher is an agent of change. The author says that these people are agents of change because they are able to arm the students with the right kind of knowledge to face the global village. A person who knows how to speak a language other than his native tongue is able to communicate more effectively. And by doing so, they are also able to spur changes in their surroundings. Suppose that a foreign exchange student is well adept at using the language of his/her host country. That student is empowered to make a difference in the lives of his host family and host institution by sharing his thoughts and ideals through the language used in that place. And he/she will only be able to do so if his/her foreign language teacher was able to act as an agent of change to the student. As a final word, different teachers can assume different roles in different situations. It may be that at one point a teacher is task-oriented (meaning the teacher is focused on how each activity or lesson will improve the language skills of the students). Another is that the teacher can also be a counselor or a colleague which are both supportive in nature. Their mindset is to join the students along the way and help them as if they are both on the same ground. Lastly, the motivational teacher is focused on how he will be able to create more value out of the student’s learning experience. Often times, the motivational teacher does not teach the language concepts in isolation with the benefits and necessity of learning such. These two ideas are intertwined to create more significance for every lesson. As of today, we are still on the lookout for teachers who are able to balance the different facets of teaching roles. But â€Å"generally, all the teachers are preoccupied with task-related functions, treating the aspect of motivation and interpersonal roles as a useful but not an indispensable addition† (The role of the foreign language teacher in the classroom — the teachers† beliefs and role behavior versus the students† expectations. , 2002) Conclusion Ultimately, language will be an important part of our lives. No matter what differences will come about in the upcoming years it will continue to become the medium by which people are to express their thoughts and ideas. Moreover, this will significantly increase the need for people to use language in nation-building purposes and for the betterment of the society where they belong to. Bibliography Anderson, S. R. (2005, April). How Many Languages are There in the World? Retrieved July 10, 2010, from Linguistics Society of America: http://www. lsadc. org/info/pdf_files/howmany. pdf Andrea A. Lunsford, j. J. (2001). Everything’s an Argument. Massachusetts: Bedford St. Martin’s. Bourdieu, P. (1977). Cultural Reproduction and Social Reproduction. New York: Oxford University Press. Davis, W. (1952). Practical World Language. The Science News-letter , 10-11. Met, M. (n. d. ). Improving Student’s capacity on Learning Foreign Language. Phi Delta Kappan , 214-218. Mulford, G. W. (1989). Semantic Processing for Communicative Exercises in Foreign-Language Learning . Computers and the Humanities , 31-44. Rampton, B. (2002). Ritual and Foreign Language Practices at School . Language in Society , 491-525. Stephen W, L. K. (2008). Theories of Human Communication. California: Thomson Wadsworth. Stevens, G. (1999). Age at Immigration and Second Language Proficiency among Foreign-Born Adults. Language in Society , 555-578. Stewart, B. (2010, March 13). Marshall McLuhan Foresees The Global Village. Retrieved July 9, 2010, from Living Internet: http://www. livinginternet. com/i/ii_mcluhan. htm Tender, A. M. (2008). About The Linguistic Constitution of Social Space: The Case of Estonia. TRAMES , 151-182. The role of the foreign language teacher in the classroom — the teachers† beliefs and role behavior versus the students† expectations. (2002). Studia Anglica Posnaniensia: international review of English Studies . Urciuoli, B. (1995). Language and Borders. Annual Review of Anthropology , 525-546. What is Language. (2006, August 31). Retrieved July 9, 2010, from Language and Culture: http://anthro. palomar. edu/language/language_2. htm What is the Importance of Learning a Second Language? (n. d. ).

A History of Happiness

The concept of happiness is one that causes controversies, some says it doesn’t exist and others revere it. In this particular case one can read the facts of this phenomenon on the article A History of Happiness. In this article is recounted how the view of happiness has changed with time. Furthermore, it is compared one against the other. It is mentioned how linguistics shows the happiness as the human kind seeks being well and how this relates to our concept of it. â€Å"Every Indo-European language without exception going way back to ancient Greek, the word for happiness is associated with the word luck†.It is quite credible this assumption as the author gives examples. This leads one to appreciate the way the ancient viewed happiness. For them it was given not obtained, it â€Å"was what happened to us†. Also it was believed blindly that it was utterly out of human hands. Therefore, the Classical philosophy main aim was concentrated on happiness and some of t he most known ancient philosophers focused on it. Although, their view was that happiness could be earned. Moreover, it was thought that it was â€Å"an outcome of moral comportment† rather than an emotional state. So only a few would be able to achieve it.As an antithesis of the ancient view, the modern type of happiness was transformed as right. Thomas Jefferson declares it to be a â€Å"self-evident truth† while George Mason speaks of the chase and obtain of happiness as a â€Å"natural endowment†. In this manner, gives a democratic view of the concept. Which lead to an open abhorrent of any kind of oligarchy towards what it should be had by mankind. Then we have John Locke, an English philosopher, who said that human â€Å"shouldn’t assume that suffering is our natural lot†. Also, that one should maximize one’s pleasure on Earth. Hence we acquire the belief of feeling good above being good.There’s a darker side to everything and h appiness is not the exception. This new perspective make people believe that it can be acquired and â€Å"consumed† rather a well lived life. In this way it makes one think that it should be had all the time without any kind of effort. However science’s discoveries tell otherwise, they believe that some of the ancient views should be kept. This serves as a counter to the obsession the right of happiness has brought. People think if they’re not in happiness they are abnormal and it leads them to search for it in a non- materialistic happiness.Therefore it’s no wonder church’s are replete. The newly acquired knowledge serves me to further explain my standpoint about the phenomenon. People become slaves of an ideal which it utterly nuts and in the end human don’t evolve from emotional frame of mind and therefore doesn’t invite to critical thinking. I found this excerpt quite interesting, by that I mean it left me unsatisfied. It served i ts purpose, to leave unsatisfied and thirsty the reader. I admire his courage for actually speaking of the unspeakable, as it dares one to actually entails our beliefs and actually approach the whys of it.