Plea Bargaining Essay Example for Free

Plea Bargaining Essay Search for justice is a sensitive process. It entails accurateness and reliability. The truth, which is the road to justice, must fully be established without any trace of doubt. The process itself is complicated and time- consuming. Nonetheless, it has to be undergone by every party because of the precious life that is at stake. In criminal prosecution, the defendant has been guaranteed rights and privileges. This is so not because the legal system favors him but to shield himself from any arbitrary manipulation of the court or any legal officer. Constitutionally, the accused is presumed innocent unless proven beyond reasonable doubt (Davis, 2007, p. 44). This is a constitutional grant that cannot be surrendered by the accused. Most importantly, it is to avoid incrimination of innocent people. During the prosecution of the case, it is also very important that the accused undergoes the process of plea bargaining. Generally, plea bargaining is â€Å"the negotiation of an agreement between a prosecutor and a defendant whereby the defendant is permitted to plead guilty to a reduced charge† (Merriam- Webster Online, 2008). Specifically, from the word bargaining, it entails two or more parties, usually the defendant and the prosecutor. On the bargaining, the defendant is given a choice whether to plead guilty and waive his constitutional right or to continue to trial. But before the defendant makes his choice, the prosecutor presents the circumstantial evidences that would lead to the conviction of the defendant. Sometimes, the prosecutor offers dropping some charges against the defendant or the crime will be lowered to a lighter offense in exchange of guilty plea by the defendant. It can be observed here that the prosecution is in control of the plea bargaining (Davis, 2007, p. 43). The plea bargaining process usually evokes negative reactions because of the belief that the defendants right is curtailed (Davis, 2007, p. 43). However, in many instances, it offers several benefits for both the defendant and the prosecutor. It has been said that criminal prosecution consumes time, money, and effort. Plea bargaining offers the shortening of the trial process when the defendant pleaded guilty. On the part of the prosecutor, if the defendant pleaded guilty, he no longer needs to conduct trials in every case that he prosecute (Davis, 2007, p. 43). Notably, the burden of proving guilt beyond reasonable doubt lies on the prosecutor. If the defendant pleaded guilty, he is no longer required to establish every element of the crime being attributed to the defendant. In addition, the prosecutor will be freed from stress of for presenting witnesses, sending subpoenas, preparing statements for every testimonies, and argue on every issue. Thus, the trial is shortened and the prosecutor can have other time for his other duties. However, in order to obtain all these, the prosecutor must offer an encouraging and strong incentive to the defendant. Th defendants rights to trial and innocence unless proven are a very essential privileges that cannot easily be surrendered. However, when the prosecution can easily establish the elements of the crime committed, the burden will certainly be met and the defendant will be put to prison. In addition, if the prosecutions evidences are strong enough, conviction for the defendant will easily be achieved. In these case, it is advantageous for the defendant to plead guilty so as not to expose him to long and tiring trial period. Additionally, it will also shun his family away from stressful trial process. Most importantly, some charges will be dropped in his favor, and he will also be penalized for lesser offense. It also follows that his criminal liability or term of imprisonment will be shortened. Furthermore, plea bargaining do not necessarily sacrifice the rights of the defendant because at the onset of the plea bargaining, he has given the freedom whether to enter a plea or not. The defendant is very aware of the consequences that the trial may bring to his life and his family, and at this particular stage, he is given a choice to obtain a favorable, or mitigating grant for himself. But definitely, if he is strongly aware that he is innocent, then his rights to trial and presumption of innocence shall be granted.

Capital Punishment :: essays research papers

Capital Punishment - History Capital Punishment - History There have been many controversies in the history of the United States, ranging from abortion, gun control, but capital punishment has been one of the most contested issues in recent decades. Capital punishment is the legal infliction of the death penalty on persons convicted of a crime. It is not intended to inflict any physical pain or any torture; it is only another form of punishment. It is irrevocable because it removes those punished from society forever, instead of temporarily imprisoning them. The usual alternative to the death penalty is life-imprisonment. Capital punishment is a method of punishment as old as civilization itself. The death penalty has been imposed throughout history for many crimes, ranging from treason to petty theft and murder. Many ancient societies accepted the idea that certain crimes deserved capital punishment. Ancient Roman and Mosaic law endorsed the notion of retaliation; they believed in the rule of "an eye for an eye." Similarly, the ancient Egyptians all executed citizens for a variety of crimes. The most famous people to be executed is Jesus. Only in England, during the reign of William the Conqueror was the death penalty not used, although the results of interrogation and torture were often fatal. Later, Britain reinstated the death penalty and brought it to its American colonies. Although the death was widely accepted throughout the early United States, not everyone approved of it. In the late-eighteen century, opposition to the death penalty gathered enough strength to lead to important restrictions on the use of the death penalty in several northern states, while in the United States, Michigan, Wisconsin, and Rhode Island abandoned the practice of it altogether. In 1794, Pennsylvan ia adopted a law to show the different degrees of murder and only used the death penalty for premeditated first-degree murder. Another reform took place in 1846 in Louisiana. This state abolished the mandatory death penalty and authorized the option of sentencing a capital offender to life imprisonment rather than to death. After the 1830s, public executions ceased to be demonstrated but did not completely stop until after 1936. Throughout history, governments have been extremely inventive in making ways to execute people. Executions inflicted in the past are now regarded today as ghastly, barbaric, and unthinkable and are forbidden by law almost everywhere. Common historical methods of execution included: stoning, crucifixion, burning, breaking on the wheel, garroting, beheading or decapitation, shooting and hanging .

Facebook Stalkers

Gabriel Montelongo Mr. Segura English 1301. 001 18, February 2013 The Book of All Evil There is no denying that social networking is at its finest. With over 1 billion users, Facebook has pushed itself into part of everyday lives. However, Facebook isn’t all flowers and butterflies. A website that stalkers use to rub one off to whoever they stalk, should not be perceived as a good thing. Neither should it be considered a good thing when a college student tries to write his essay, and keeps getting distracted by notifications.Facebook is a tool used for stalking, addiction, but worst of all, procrastination. One of the scariest thing about Facebook is the stalkers. Facebook has made stalking easier with the use of one of its features called â€Å"Lists†. Basically what this feature does is it allows the user to single out anyone in the friends list, and stalk anyone that has been added. The stalker can even single out what kind of updates a person has; such as photos, po sts, and recent locations visited.People that use Facebook should think twice about what they share with the world. On the bright-side, Facebook does allow the ability to block stalkers, but it is rendered useless if the stalker decides to fake a new account. If anyone out there has an interest in stalking, Facebook would be the first place they would look. When awkward, socially-undeveloped, cave-dwellers want to make friends, they use Facebook. They proceed to add pretty, young women, in hopes that they get a chance to be with them.However, that almost never happens, so these people then become angry and delusional. This is how stalkers tend to form over Facebook. These stalkers then proceed to stalk their rejecters, and then later end up in creepy stalker stories. This happens almost every day on Facebook, seriously. Social networking has become so common, that not many think about how they are being manipulated. Facebook has this control over people, using something called †Å"notifications†.People post updates on their page, and if people think it’s cool, then it’ll get â€Å"likes†. Notifications are then sent to the person telling them that someone liked their update. That right there encourages the user to post more updates. The more likes that people receive, the more it makes them want to update their status. This is how it keeps its users; through making them think that people actually care about what they are saying. Facebook manages to keep its users through a clever, but mischievous design.Though widely recognized as a place for friends, Facebook is more like a spreading of disease. Through mischievous design and manipulation, Facebook can actually cause addiction. A reward system of notifications and likes can make a person feel like something was accomplished through something as simple as a status update. This accomplishing feeling make the self-conscious Facebook users feel accepted through having their statuses â€Å"liked†, and will try to reanimate themselves in search of that same feeling.Facebook addiction is now being recognized as a legitimate addiction, and will soon find itself in future psychology books. Not only is Facebook perfectly designed for stalkers, but also to distract students from finishing homework. Facebook has a billion plus users, a majority of which are teenagers and young adults. Distracting a kid couldn’t be easier with Facebook. Give a pre-teen a website full of friends, and that pre-teen could be on it for hours.The same effect happens with young college students; only the college student knows that he has an essay due Wednesday. Said student will then proceed to write the essay, only to get distracted by friends on Facebook. It is a horrible feeling when students know that not much time is left, and begin to hate themselves for procrastinating. An evil that every teenager in this day and age has to deal with, every day of their lives. Facebook is a curse; one that destroys us from the inside out.Never mind the fact that almost one in every five divorces have been Facebook related, it creates stalkers that could eventually evolve into serial killers. Facebook breeds addiction and manipulates our already screwed up society. It has students thinking that they will only be on for five minutes, which then becomes an hour. Facebook is itself a problem that causes even more problems. A problem that’s here to stay until a new socializing website comes and takes over, and causes even more problems.

Terrorism Threat In South Asia And Its Implications International Law Essay - Free Essay Example

Sample details Pages: 9 Words: 2652 Downloads: 1 Date added: 2017/06/26 Category Law Essay Type Research paper Did you like this example? Almost no week goes by when in India innocent lives are not lost to terrorist attacks. The horrific terrorist attack on Mumbai in November 2008 has underscored the nature of the terrorist threat in South Asia. The spread of terrorism in South Asian region has been rampant over the last few decades. Don’t waste time! Our writers will create an original "Terrorism Threat In South Asia And Its Implications International Law Essay" essay for you Create order While India battles with the terrorism in Jammu and Kashmir to insurgency in North East and the Maoist attacks in central India the others in the region are not left untouched. For eg Nepal has seen the spread of Maoists beyond control, Sri Lanka faced the worst kind of insurgency and finally overcame but at a huge cost to human lives, while Pakistan, which encouraged increased fundamentalism and militancy in the tribal areas, is now becoming a target of its homegrown terrorist movements.  [1]  The list of heinous terrorist attacks in different countries of the region is endless while the region is turning into the breeding ground for terrorist due to poverty, poor governance and geopolitical tensions which are peculiar to the region. Therefore it is imperative to tackle the problem at the regional level in order to find any lasting solution to the problem. Diverse Categories of Terrorist Groups. South Asia has been a victim of violence perpetrated by a myriad of groups with d iverse objectives and varied ideologies, which can be categorized as under  [2]  :- Nationalists. These are those motivated by nationalist ideologies. The most prominent of these have been ones focused on the political future of Kashmir in India.  [3]  Most of the violence can be attributed to three groups, viz, Hizbul Mujahideen, Lashkar-e-Toiba (LeT), and Jaish-e-Mohammad (JEM). Of the three, the latter two, LeT and JEM draw most of their recruits from Pakistan. Religious Extremists  [4]  . The main geographical area in which this is prevalent is the Afghan-Pakistan border. Al Qaida which continues to threaten the peace and stability in the world at large is believed conducting its activities through this region. Although the core centre of religious extremism is pointed at Pakistan and Afghanistan by the world community but it spread to other countries like Bangladesh is very much a reality which is proved by a spate of terrorist attack from there. The convergen ce of religious fundamentalist along with organized crime groups, national or transnational, thus adds additional dimension of complexity to the terrorist threat in the region. Ethno-Nationalist Separatists. Separatist groups are active in every country in the subcontinent. The past struggle between the LTTE and the government of Sri Lanka remains one of the bloodiest insurgencies in South Asia. Separatist movements are also present in Pakistans Balochistan and Sindh provinces, Indias Jammu and Kashmir and Northeastern provinces, and Bangladeshs Chittagong Hill Tracts. Although most of the violence in each of these conflicts is internal, it has the potential to spill over into neighboring provinces. Cross-border ethnic sub-nationalism is one of the main sources of mistrust on the subcontinent.  [5] Failure of Regional Mechanism to Counter Threat The counterterrorism efforts of regional bodies, such as the South Asian Association for Regional Cooperation (SAARC) and the Bay of Bengal Initiative for Multi-Sectoral Technical and Economic Cooperation (BIMSTEC), have often been marginalised due to mistrust among the states. Although both bodies have devised potentially useful counterterrorism instruments, few such measures have been translated into action by the member states. Measures Initiated by SAARC. SAARC has been ahead of many regional bodies in the adoption of legal instruments for fight against terrorism. Its regional convention on the suppression of terrorism in 1987 includes a definition of terrorist acts and calls for greater regional cooperation on legal issues, including evidence sharing, extradition, and information and expertise exchange.  [6]  In an additional Protocol updated in 2002 it incorporated into the original convention the obligations of states under UN Security Council Resolution 1373, adopted in the aftermath of 9/11. A not very encouraging sign in the protocol is its focus on law-enforcement or hard security measures. To further the implementation of the provisions in its convention, SAARC created the Terrorist Offences Monitoring Desk (STOMD) in Colombo, Sri Lanka. However despite the appearance of counterterrorism on the agenda and the creation of STOMD, there has been little forward movement beyond the rhetorical level. There are widely held suspicions that the intelligence agencies of various South Asian states have designs on their neighbours territories, or facilitate insurgent movements to entrench political rivals in asymmetric warfare, and this further fuels the reluctance to share information and resources.  [7]  Due such suspicions and tensions among member states few of the counterterrorism instruments and commitments adopted by SAARC in its more than twenty year history have been translated into action by its members. One recent example is the apparent unwillingness of Pakistan to extradite to India those suspected of involvement in the 2008 Mumbai attacks, despite the existence of these instruments. Importance of BIMSTEC. The other important organization in the region to coordinate the fight against terrorism is the Bay of Bengal Initiative for Multi-Sector Technical and Economic Cooperation (BIMSTEC) which includes Bangladesh, Bhutan, India, Myanmar, Nepal, Sri Lanka, and Thailand. It was founded in 1997 with an aim to facilitate cooperation in areas such as trade and investment, technology, energy, transportation, communications, and tourism. In its 2004 Summit Declaration BIMSTEC expressed concern about the threat of terrorism to regional trade and urged all member states to coordinate their efforts by exchanging information and cooperating in the ongoing efforts of the international community to combat terrorism in all its forms. In order to coordinate the sub regional effort a Counter Terrorism and Transnational Crime Se ctor (CTTCS) was established. Though yet to establish a headquarters or permanent Secretariat, BIMSTEC has been slow to get many of its programs off the ground. The real weakness or strength depending on the perspective lies in the absence of Pakistan from the association. While this makes easier to forge consensus, but limits the region wide promotion of counter terrorism cooperation in the region. Relevance of Counter Terrorism Strategy in South Asia While any meaningful progress within SAARC has been held hostage to the tensions between South Asias two largest rivals, BIMSTEC faces limitations, both in terms of its capacity as an organization and its utility in promoting region-wide counterterrorism cooperation because Pakistan is not a member.  [8]  Therefore the limitations of SAARC and BIMSTEC point to the importance of the United Nations and its role in promoting counterterrorism cooperation and capacity-building activities in the region in the framework of the United Nations Counter Terrorism Strategy. This provides an opportunity for United Nations to shape a regional response to terrorism, using the holistic United Nations Strategy as an entry point for enhanced engagement in the region. The terrorist attacks in Mumbai in November 2008 has made it very clear that strengthening counterterrorism cooperation in South Asia must be a top priority as it is not only a threat to national security, but to regional stability and international peace as well. Subsequent paragraphs highlight the relevance of each pillar of the Strategy for South Asia and the role that the different organizations of the United Nations system, many of which now form the entities of CTITF, can play.  [9] Pillar I: Measures to Address Conditions Conducive to the Spread of Terrorism. While these objectives of the Pillar 1 are long term, the strategy does highlight some specific ways in which different elements of the United Nations system can contribute towards the fulfilment of the Pillar I objectives which are enumerated in subsequent paragraphs.  [10] United Nations initiative to promote cross-cultural understanding through UNESCO and enhance inter and intra faith dialogue and dialogue among different communities, the Strategy encourages the United Nations system as a whole to increase cooperation and assistance in the fields of law, human rights and good governance, to support sustained economic and social develop ment. United Nations Development Programme (UNDP) has made an effort to address inequities in development and access to resources thereby improving governance and respect for the rule of law in South Asia. It has contributed to undermining the chronic poverty, inequality, and social injustices which fuel much of the political violence in the region. United Nations has played an important role in conflict prevention and crisis management. In South Asia, the United Nations has missions in Nepal and Afghanistan and a military observer group deployed along the Line of Control in India and Pakistan. The UN Mission in Nepal (UNMIN) provides an example of where the United Nations has played a largely constructive role in monitoring ceasefire arrangements and providing technical support for the conduct of the election of a Constituent Assembly in a free and fair atmosphere. Pillar II : Measures to Prevent and Combat Terrorism. The second pillar of the United Nations Strategy includ es a series of short-term, preventative measures that states are to take to address the terrorist threat. While all states in the region have taken some steps to implement the measures elaborated in the Strategys second pillar, particularly when it comes to enhancing national criminal justice and other law-enforcement responses. Examples include the adoption of an anti-money-laundering (AML) ordinance in Pakistan by presidential decree in September 2007, the enactment of a range of counterterrorism laws and the establishment of a Financial Intelligence Unit (FIU) in Sri Lanka and the computerization of some checkpoints along Indias borders. Bangladesh has also taken steps to guard against terrorist financing under AML legislation. Counter-Terrorism Committee (CTC) and its Counter-Terrorism Executive Directorate (CTED), which play an important role in furthering the implementation of this pillar have so far visited three countries in South Asia i.e Bangladesh, India, and Pakistan. Du ring these visits it has not only discussed law enforcement, border, and other security-related issues, but issues related to the prevention of radicalization and extremism such as school curricula and other education reforms, pursuant to the part of its mandate derived from Security Council Resolution 1624 which calls upon states to take measures to prevent radicalization and incitement to terrorism and promote dialogue among cultures and religions.  [11] Pillar III: Measures to Build States Capacity to Prevent and Combat Terrorism and to Strengthen the Role of the United Nations in this Regard. South Asia confronts enormous development challenges including widespread poverty and over population. Therefore the region faces capacity challenges in trying to implement the United Nations Strategy as well as in developing and implementing national counterterrorism strategies and initiatives. The ability of countries in the region to implement the Strategy is further compromised by underdevelopment, especially in the border areas, as well as poor governance, unemployment, corruption, and the lack of trust and limited cross-border cooperation. More generally, many South Asian states have strong central governments, but are weak at the local levels and poor at delivering goods and services to their populations. Thus, for example, law enforcement and other criminal justice officials may be very capable at the national level, significant shortfalls exist at the local level, in terms of numbers, training, and equipment. Pillar III of the Strategy is thus of critical importance for South Asia as efforts are made to translate the holistic UN Strategy into action on the ground.  [12] Pillar IV: Measures to Ensure Respect for Human Rights for all and the Rule of Law as the Fundamental Basis of the Fight against Terrorism. The introduction of special or extraordinary laws to address terrorism can be particularly detrimental to the protection of human rights, as they can lead to long term institutionalization of oppression and foster a culture of impunity within state security forces and agencies. In India, for example, the government repealed the 2002 Prevention of Terrorism Act (POTA) to institute more humane reforms. In Pakistan, Suppression of Terrorist Activities Ordinance, 1975, was challenged on human-rights grounds. The Act was then repealed and replaced by the Anti-terrorism Act of 1997, which, among other things, created special antiterrorist courts. In the light of capacity and political challenges and the lack of an effective intergovernmental human rights mechanism on the subcontinent, the role of the United Nations system is particularly important. This makes the OHCHRs stated plans to establish a regional office in South Asia of particular significance. Among the countries that would be covered by this office are Afghanistan, Bangladesh, Bhutan, India, the Maldives, Nepal, and Sri Lanka.  [13] The Way Ahead While it is clear the key role that United Nations is playing to shape a regional response against terrorism in the South Asian region but there is still a huge scope to foster peace and stability in the region. There are certain recommendations that the global body needs to institute in order to link the counter terrorism strategy more closely to the needs of the South Asian region. South Asian region requires a more tailor made organisation to fight terrorism considering the political and regional complexities of the region. Since there is very little cooperation among the states and certain states even sponsor terrorism to promote their aspirations there is need for more coercive policy of the Security Council against states which still promote fundamentalism and militancy to account for their actions under Chapter VII. United Nations should build on increasing recognition of the importance of combating terrorism, conflict, and political violence in South Asia to forge str onger cooperation on the implementation of the Global Counter-Terrorism Strategy between the United Nations and South Asia, and within the region itself. SAARC leaders should provide full support to Strategy and initiate a approach to implement measures proposed by the Strategy to counter the conditions conducive to terrorism, prevent and combat terrorism, build state capacity, and streamline a human rights approach to all counterterrorism measures. Greater consideration should be given to conducting UN counterterrorism country visits which should probably use the General Assemblys UN Strategy as a framework rather Chapter VII mandate. Such an approach might allow for more holistic United Nations engagement with South Asia on counterterrorism issues, but lower the political temperature of the visit and thus enhance its technical focus. UNODCs Terrorism Prevention Branch should seek to ramp up the provision of technical assistance to criminal justice officials in South Asia regarding the ratification and implementation of the sixteen international conventions and protocols related to terrorism. For example, it could seek to bring together criminal justice practitioners from all countries in the region for common training sessions that would allow for the expert to expert contacts that are essential to building cross-border trust. A significant step toward protection human rights while countering terrorism in South Asia should be an OHCHR organized seminar that brings together not only representatives of foreign ministries, national human rights institutions, and nongovernmental organizations from countries in the region, but also security and law-enforcement officials from throughout South Asia. Such a forum would provide an excellent opportunity for government and nongovernment experts from the region to exchange experiences, challenges, and best practices in addressing the common terrorist threat in a manner that is consistent with human rights no rms.  [14] Summary Given the complexity of the threat, the geopolitical animosities on the subcontinent, and the development challenges in the region, South Asia region is an apt place for terrorist groups to further their cause without difficulty. There is a strong need for a holistic approach to address the terrorist threats confronting the region which includes includes both hard and soft and short and long term measures, to combat a transnational threat and emergent non state actors. In the absence of a meaningful regional framework to facilitate the cooperation necessary to respond to and prevent future terrorist acts there is a need to develop an effective regional counter terrorism response. Consequently, it can be argued that the United Nations is well placed to stimulate greater regional cooperation on counter terrorism, especially given its comparative advantages as a result of its neutrality, expertise, and stature in the global world. Having analysed all aspects of the UN Global Counte r Terrorism Strategy and its relevance in the South Asian region, there is requirement to list out certain logical recommendations in light of the shortcomings observed in the strategy.

Out, Out by Robert Frost - 782 Words

Robert Frosts poem â€Å"Out, Out,† paints a strange and bizarre death image to readers; A young boys death due to a carnivorous chainsaw who sought blood, slicing the boys hand off. Robert makes readers understand why he would paint such a tragic accident with various narrative elements, such as personification, many signs of imagery, emotions, and perceptions throughout the story. Also, Frost references William Shakespeare’s work, â€Å"Macbeth.† This gives readers who have read Macbeth before, an idea of what’s to come in the end of the poem, the feeling of sadness and death. This analysis will show the main theme of the boys death, who died doing the work of a man. Robert Frost starts out by showing readers the year and location; 1945, New England, Vermont. This is the period where World War I occurs, however, Frost does not describe the land destroyed and covered with dead bodies, instead Vermont is a place where exists; â€Å"Five Mountain ranges on e behind the other under the sunset far into Vermont.† This makes readers understand that this place is located in a beautiful place out in the wilderness away from war. The location is very significant to the story, I have done a bit of research about Frosts life; Frost was forced to return to America due to the war in 1915, a war that would have killed hundreds of innocent little boys. With this being said, the reader now understands why the boy in the poem is located in a place like Vermont in a time of war, and why the boy in theShow MoreRelatedOut Out by Robert Frost548 Words   |  2 PagesRobert Frost is the author of Out Out--, â€Å"Stopping by Woods on a Snowy Evening, and Nothing Gold can Stay. His literary work communicates deep meaning through the use of metaphoric language and deception. Being raised most of his life on a farm; his works perceive the natural life of a normal person while out in nature. â€Å"Frost believes that the emphasis on everyday life al lows him to communicate with his readers more clearly; they can empathize with the struggles and emotions that are expressed inRead MoreAnalysis Of Out, Out By Robert Frost727 Words   |  3 PagesIn Robert Frost’s poem â€Å"Out, Out† an overwhelming theme of agony can be sensed as Frost incorporates his personal experiences with loss and his views on society into the narrative of this literary work. Frost uses the depiction of innocence through a young boy who suffers a fatal accident to metaphorically embed his personal struggles with the death of his two children into the poem. The section of the poem that will be analyzed is the final ten lines (25-34). The significance of this section inRead MoreAnalysis Of Out, Out By Robert Frost780 Words   |  4 PagesOut, Out and the Responsibilities of age Responsibilities may not seem very harmful, but not adhering to these responsibilities can lead to dire consequences. Said responsibilites are much more prominent in the teenage years of life. For example: driving, getting offered drugs and/or alcohol, and intercourse are all situations that teenagers might find themselves in. These situations may not be inherently bad, but because teenagers are new drivers, too young and inexperienced to properly care forRead MoreOut, Out by Robert Frost Essay836 Words   |  4 PagesOut, Out Out, Out, by Robert Frost is a gruesomely graphic and emotional poem about the tragic end of a young boys life. It is a powerful expression about the fragility of life and the fact that death can come at any time. Death is always devastating, but it is even more so when the victim is just a young boy. The fact that the boys death came right before he could Call it a day (750) leads one to think the tragedy might have been avoided and there by forces the reader to think, WhatRead MorePoem, Birches And Out, Out By Robert Frost1116 Words   |  5 PagesTheme, Figurative Speech and Tones in â€Å"Birches† and â€Å"Out, Out† by Robert Frost Robert Frost was born in 1874 in San Francisco. Descended from the New Englanders generations, his parents, make Robert Frost is much associated with New England. In addition, most of his poems were well-known as a reflection from New England life. Despite that, he was a kind of subtle poet and generally recognized as a private man. Moreover, his appearance at the inauguration of John F. Kennedy to recite â€Å"The Gift Outright†Read More Analysis of Out, Out by Robert Frost Essay591 Words   |  3 PagesAnalysis of Out, Out by Robert Frost Robert Frost tells a disturbing story in Out, Out, --, in which a little boy loses his life. The title of the poem leaves the reader to substitute the last word of the title, which some would assume would be out because of the repetition. The title is referring to the boy exiting the living world. Frost drags the readers mind into the poem with the imagistic description of the tools and atmosphere the little boy is surrounded by. Frost describesRead More Appeal of Robert Frosts Out Out Essay1055 Words   |  5 Pagesinteresting and appealing poems is Robert Frost’s â€Å"Out, Out†. The poem has the ability to make the reader visualize an event in vivid detail without making it into a short story. The poem depicts a very dramatic scene and makes it seem as if the reader is really there. Poems are generally thought to be about love and feelings, but some poems can actually be like a short story; these are called narrative poems, which means that they tell a story. The poem â€Å"Out, Out† is a great example of a narrativeRead MoreLooking Out By Robert Frost2826 Words   |  12 Pageshas found its thought and the thought has found words.† (Robert Frost) Expressing emotions is a very important thing that we do in our lives, everyone also has a different way they express emotions. In the quote by Robert Frost he says that he expresses his emotions through writing poetry. The way people express emotions is very important in their lives, it can improve or destroy many interpersonal relationships. In the book â€Å" Looking Out, Looking in† it talks about how people express emotions toRead MoreDisabled by Wilfred Owen and Out, Out by Robert Frost1516 Words   |  6 PagesCompare how the theme of loss is communicated in the poems â€Å"Disabled† by Wilfred Owen and ‘Out, Out –‘by Robert Frost In both of the poems â€Å"Out, Out’’ and ‘’Disabled’ ’has a similar theme of loss and is shown throughout each poem. Both of the poem deals with the subject of physical loss. The characters of these poems both experience losses from an accident. They create an effect, where the audience will show empathy to the two poems. In order to create this outstanding effect, they both used similarRead More A Comparison of The death of a hired man and Out, Out- by Robert Frost1199 Words   |  5 PagesA Comparison of The death of a hired man and Out, Out- by Robert Frost Robert frost was born in Vermont in 1874 and died in 1963. Robert Frost was a farmer and lived in Vermont, USA. Both poems The death of a hired man and Out, Out- are set on a farm in Vermont which is probably because of where Robert Frost lived and worked. I will know begin to discuss the similarities. As I said previously both poems are set in a farm enviroment. The poem The death of a hired man is probably set

Using Math to Find a Way to Beat the Game A Knight´s Tour

I chose this specific topic because I was interested to know if there was an actual mathematical way of beating the game called â€Å"A Knight’s Tour†. Whenl was first introduced to this game in my research for an internal assessment topic, I was rather intrigued by its goal that was required for each player to attempt to achieve. I thought that the game wouldn’t be too terribly hard because I knew fluently how the knight moved and thought that I would be able to move it in certain patterns across the board that would allow me to complete the knight’s tour. Although the goal of the game seemed easily obtainable at first glance, as I continued to play the game and take myself on a lnight’s tour, I came to conclusion that it would be nearly impossible to beat without having an algorithm to base each move off of. With each attempt at the game, I try different movements that might help to beat â€Å"a Knight’s Tour† but always end up landi ng on the same square more than once because I run out of moves, or failing to beat the game in the required 63 moves or less. 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Computational Efficiency of Polar Free Essays

string(127) " the n component multivariate normal with D = I , just take the components of Y to be independent univariate standard normals\." Lecture Notes on Monte Carlo Methods Fall Semester, 2005 Courant Institute of Mathematical Sciences, NYU Jonathan Goodman, goodman@cims. nyu. edu Chapter 2: Simple Sampling of Gaussians. We will write a custom essay sample on Computational Efficiency of Polar or any similar topic only for you Order Now created August 26, 2005 Generating univariate or multivariate Gaussian random variables is simple and fast. There should be no reason ever to use approximate methods based, for example, on the Central limit theorem. 1 Box Muller It would be nice to get a standard normal from a standard uniform by inverting the distribution function, but there is no closed form formula for this distribution 2 x unction N (x) = P (X x) = v1 ? e? x /2 dx . The Box Muller method is a 2 brilliant trick to overcome this by producing two independent standard normals from two independent uniforms. It is based on the familiar trick for calculating ? 2 e? x I= /2 dx . This cannot be calculated by â€Å"integration† – the inde? nite integral does not have an algebraic expression in terms of elementary functions (exponentials, logs, trig functions). However, ? 2 e? x I2 = ? /2 e? y dx 2 ? /2 ? 2 e? (x dy = +y 2 )/2 dxdy . The last integral can be calculated using polar coordinates x = r cos (? ), y = r sin(? with area element dxdy = rdrd? , so that 2? I2 = r = 0? e? r 2 /2 rdrd? = 2? r = 0? e? r 2 /2 rdr . ? =0 Unlike the original x integral, this r integral is elementary. The substitution s = r2 /2 gives ds = rdr and ? e? s ds = 2? . I 2 = 2? s=0 The Box Muller algorithm is a probabilistic interpretation of this trick. If (X, Y ) is a pair of independent standard normals, then the probability density is a product: 2 2 1 1 ? (x2 +y2 )/2 1 e . f (x, y ) = v e? x /2  · v e? y /2 = 2? 2? 2? 1 Since this density is radially symmetric, it is natural to consider the polar coordinate random variables (R, ? de? ned by 0 ? ? 2? and X = R cos(? ), and Y = R sin(? ). Clearly ? is uniformly distributed in the interval [0, 2? ] and may be sampled using ? = 2? U1 . Unlike the original distribution function N (x), there is a simple expression for the R distribution function: 2? r G(R) = P (R ? r) = r =0 ?=0 r 1 ? r 2 /2 e rdrd? = 2? e? r 2 /2 rdr . r =0 The same change of variable r 2 /2 = s, r dr = ds (so that r = r when s = r2 /2) allows us to calculate r 2 /2 e? s dx = 1 ? e? r G(r) = 2 /2 . s=0 Therefore, we may sample R by solving the distribution function equation1 G(R) = 1 ? e? R 2 /2 = 1 ? U2 , whose solution is R = ? 2 ln(U2 ). Altogether, the Box Muller method takes independent standard uniform random variables U1 and U2 and produces independent standard normals X and Y using the formulas ? = 2? U1 , R = ?2 ln(U2 ) , X = R cos(? ) , Y = R sin(? ) . (1) It may seem odd that X and Y in (13) are independent given that they use the same R and ?. Not only does our algebra shows that this is true, but we can test the independence computationally, and it will be con? rmed. Part of this method was generating a point â€Å"at random† on the unit circle. We suggested doing this by choosing ? niformly in the interval [0, 2? ] then taking the point on the circle to be (cos(? ), sin(? )). This has the possible drawback that the computer must evaluate the sine and cosine functions. Another way to do this2 is to choose a point uniformly in the 2 ? 2 square ? 1 ? x ? 1, 1 ? y ? 1 then rejecting it if it falls outside the unit circle. The ? rst accepted point will be uniformly distributed in the unit disk x2 + y 2 ? 1, so its angle will be random and uniformly distributed. The ? nal step is to get a point on the unit circle x2 + y 2 = 1 by dividing by the length. The methods have equal accuracy (both are exact in exact arithmetic). What distinguishes them is computer performance (a topic discussed more in a later lecture, hopefully). The rejection method, with an acceptance probability ? ? 4 78%, seems e? cient, but rejection can break the instruction pipeline and slow a computation by a factor of ten. Also, the square root needed to compute 1 Recall that 1 ? U2 is a standard uniform if U2 is. for example, in the dubious book Numerical Recipies. 2 Suggested, 2 the length may not be faster to evaluate than sine and cosine. Moreover, the rejection method uses two uniforms while the ? method uses just one. The method can be reversed to solve another sampling problem, generating a random point on the â€Å"unit spnere† in Rn . If we generate n independent standard normals, then the vector X = (X1 , . . . , Xn ) has all angles equally n likely (because the probability density is f (x) = v1 ? exp(? (x2 + ·  ·  ·+x2 )/2), n 1 2 which is radially symmetric. Therefore X/ X is uniformly distributed on the unit sphere, as desired. 1. 1 Other methods for univariate normals The Box Muller method is elegant and reasonably fast and is ? ne for casual omputations, but it may not be the best method for hard core users. Many software packages have native standard normal random number generators, which (if they are any good) use expertly optimized methods. There is very fast and accurate software on the web for directly inverting the normal distribution function N (x). This is particularly important for qua si Monte Carlo, which substitutes equidistributed sequences for random sequences (see a later lecture). 2 Multivariate normals An n component multivariate normal, X , is characterized by its mean  µ = E [X ] and its covariance matrix C = E [(X ?  µ)(X ?  µ)t ]. We discuss the problem of generating such an X with mean zero, since we achieve mean  µ by adding  µ to a mean zero multivariate normal. The key to generating such an X is the fact that if Y is an m component mean zero multivariate normal with covariance D and X = AY , then X is a mean zero multivariate normal with covariance t C = E X X t = E AY (AY ) = AE Y Y t At = ADAt . We know how to sample the n component multivariate normal with D = I , just take the components of Y to be independent univariate standard normals. You read "Computational Efficiency of Polar" in category "Essay examples" The formula X = AY will produce the desired covariance matrix if we ? nd A with AAt = C . A simple way to do this in practice is to use the Choleski decomposition from numerical linear algebra. This is a simple algorithm that produces a lower triangular matrix, L, so that LLt = C . It works for any positive de? nite C . In physical applications it is common that one has not C but its inverse, H . This would happen, for example, if X had the Gibbs-Boltzmann distribution with kT = 1 (it’s easy to change this) and energy 1 X t HX , and probability 2 1 density Z exp(? 1 X t HX ). In large scale physical problems it may be impracti2 cal to calculate and store the covariance matrix C = H ? though the Choleski factorization H = LLt is available. Note that3 H ? 1 = L? t L? 1 , so the choice 3 It is traditional to write L? t for the transpose of L? 1 , which also is the inverse of Lt . 3 A = L? t works. Computing X = L? t Y is the same as solving for X in the equation Y = Lt X , which is the process of back substitution in numerical linear algebra. In some applications one knows the eigenvectors of C (which also are the eigenvectors of H ), and the corresponding eigenvalues. These (either the eigenvectors or the eigenvectors and eigenvalues) sometimes are called principal com2 ponents. Let qj be the eigenvectors, normalized to be orthonormal, and ? j the corresponding eigenvalues of C , so that 2 Cqj = ? j qj , t qj qk = ? jk . t Denote the qj component of X by Zj = qj X . This is a linear function of X and t therefore Gaussian with mean zero. It’s variance (note: Zj = Zj = X t qj ) is 2 t t t 2 E [Zj ] = E [Zj  · Zj ] = qj E [XX t ]qj = qj Cqj = ? j . A similar calculation shows that Zj and Zk are uncorrelated and hence (as components of a multivariate normal) independent. Therefore, we can generate Yj as independent standard normals and sample the Zj using Zj = ? j Yj . (2) After that, we can get an X using Zj qj . X= (3) j =1 We restate this in matrix terms. Let Q be the orthogonal matrix whose columns are the orthonormal eigenvectors of C , and let ? 2 be the diagonal ma2 trix with ? j in the (j, j ) diagonal position. The eigenvalue/eigenvector relations are CQ = Q? 2 , Qt Q = I = QQt . (4) The multivariate normal vector Z = Qt X then has covariance m atrix E [ZZ t ] = E [Qt XX t Q] = Qt CQ = ? 2 . This says that the Zj , the components of Z , are 2 independent univariate normals with variances ? j . Therefore, we may sample Z by choosing its components by (14) and then reconstruct X by X = QZ , which s the same as (15). Alternatively, we can calculate, using (17) that t C = Q? 2 Qt = Q Qt = (Q? ) (Q? ) . Therefore A = Q? satis? es AAt = C and X = AY = Q? Y = QZ has covariance C if the components of Y are independent standard univariate normals or 2 the components of Z are independent univariate normals with variance ? j . 3 Brownian motion examples We illustrate these ideas for various kids of Brownian motion. Let X (t) be a Brownian motion path. Choose a ? nal time t and a time step ? t = T /n. The 4 observation times will be tj = j ? t and the observations (or observation values) will be Xj = X (tj ). These observations may be assembled into a vector X = (X1 , . . . , Xn )t . We seek to generate sample observation vectors (or observation paths). How we do this depends on the boundary conditions. The simplest case is standard Brownian motion. Specifying X (0) = 0 is a Dirichlet boundary condition at t = 0. Saying nothing about X (T ) is a free (or Neumann) condition at t = T . The joint probability density for the observation vector, f (x) = f (x1 , . . . , xn ), is found by multiplying the conditional densities. Given Xk = X (tk ), the next observation Xk+1 = X (tk + ? ) is Gaussian with mean Xk and variance ? t, so its conditional density is v 2 1 e? (xk+1 ? Xk ) /2? t . 2? ?t Multiply these together and use X0 = 0 and you ? nd (with the convention x0 = 0) f (x1 , . . . , xn ) = 3. 1 1 2? ?t n/2 exp ?1 2 ? Deltat n? 1 (xk+1 ? xk )2 . (5) k=0 The random walk method The simplest and possibly best way to generate a sample observation path, X , comes from the derivation of (1). First generate X1 = X (? t) as a mean zero v univariate normal with mean zero and variance ? t, i. e. X1 = ? tY1 . Given X1 , X2 is a univariate normal with mean X1 and variance ? , so we may v take X2 = X1 + ? tY2 , and so on. This is the random walk method. If you just want to make standard Brownian motion paths, stop here. We push on for pedigogical purposes and to develop strategies that apply to other types of Brownian motion. We describe the random walk method in terms of the matrices above, starting by identifying the matrices C and H . Examining (1) leads to ? 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= ?. .. ?t ? . . 2 ? 1 ?. ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 ? 0 .? .? .? ? ? ? ? 0? ? ? ?1 ? 1 This is a tridiagonal matrix with pattern ? 1, 2, ? except at the bottom right corner. One can calculate the covariances Cjk from the random walk representation v Xk = ? t (Y1 +  ·  ·  · + Yk ) . 5 Since the Yj are independent, we have Ckk = var(Xk ) = ? t  · k  · var(Yj ) = tk , and, supposing j k , Cjk = E [Xj Xk ] = ? tE [((Y1 +  ·  ·  · + Yj ) + (Yj +1 +  ·  ·  · + Yk ))  · (Y1 +  ·  ·  · + Yj )] = 2 ?tE (Y1 +  ·  ·  · + Yj ) = tj . These combine into the familiar formula Cjk = cov(X (tj ), X (tk )) = min(tj , tk ) . This is the same as saying that the ? 1 ?1 ? ?. ?. C = ? t ? . ? ? ? 1 matrix C is 1  ·Ã‚ ·Ã‚ · 2 2  ·Ã‚ ·Ã‚ · 2 . . . 3  ·Ã‚ ·Ã‚ · . . . 2 3  ·Ã‚ ·Ã‚ · ? 1 2? ? ? 3? .? .? .? .. . (6) The random walk method for generating X may be expresses as ? ? ? Y ? X1 1 1 0  ·Ã‚ ·Ã‚ · 01 ? ? ? ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? . ? ?.? ?.? v? ? . ? ?.? 1 0 . . ? . .? ? . ? = ? t ? 1 1 ? ? ? ? ?. . .. ? ? ? ?. . . .. ? ? ? ? 11 1  ·Ã‚ ·Ã‚ · 1 Yn Xn Thus, X = AY with ? ? 1 0  ·Ã‚ ·Ã‚ · 01 ?1 1 0  ·Ã‚ ·Ã‚ · 0 ? ? ? v? .? .? . ?1 1 1 0 .? A = ? t ? ?. . ? .. .. ?. . ? . 11 1  ·Ã‚ ·Ã‚ · 1 (7) The reader should do the matrix multiplication to check that indeed C = AAt for ( 6) and (7). Notice that H is a sparse matrix indicating short range interactions while C is full indicating long range correlations. This is true of in great number of physical applications, though it is rare to have an explicit formula for C . 6 We also can calculate the Choleski factorization of H . The reader can convince herself or himself that the Choleski factor, L, is bidiagonal, with nonzeros only on or immediately below the diagonal. However, the formulas are simpler if we reverse the order of the coordinates. Therefore we de? ne the coordinate reversed observation vector t X = (Xn , xn? 1 , . . . , Xn ) and whose covariance matrix is ? tn ? tn? 1 ? C=? . ?. . t1 tn? 1 tn? 1  ·  ·  · t1 t1 .. .  ·Ã‚ ·Ã‚ · ? ? ? , ? t1 and energy matrix ? 1 ? 1  ·Ã‚ ·Ã‚ · 0 ? 0 .? .? .? ? ? ?. ? 0? ? ? ?1 ? 2 ? ? ? 1 2 ? 1 0  ·Ã‚ ·Ã‚ · ? ? .. .. .. . . . 1 ? 0 ? 1 ? H= .. ?t ? . . ?. . 2 ? 1 ? ? .. ? . ? 1 2 0  ·Ã‚ ·Ã‚ · 0 ? 1 We seek the Choleski factorization H = LLt ? l1 0 ? m2 l2 1? L= v ? m3 ?t ? 0 ? . .. . . . with bidiagonal ?  ·Ã‚ ·Ã‚ · ? 0 ? ?. .. ? . ? .. . Multiplying out H = LLt leads to equatio ns that successively determine the lk and mk : 2 l1 l 1 m2 2 2 l1 + l 2 l 2 m3 = 1 =? l1 = 1 , = ? 1 =? m2 = ? 1 , = 2 =? l2 = 1 , = 1 =? m3 = ? 1 , etc. , The result is H = LLt with L simply ? 1 0  ·Ã‚ ·Ã‚ · ? ? 1 10 1? .. L= v ? . ?t ? ? 1 ? . .. .. . . . . 7 ? ? ? ?. ? ? The sampling algorithm using this Y = Lt X : ? ? ? 1 Yn ? Yn? 1 ? ? ? ? ?0 ? ? 1? ? ? ? ? . ?= v ? ?.? ?t ? ?.? ?. ? ? ?. . Y1 0 information is to ? nd X from Y by solving ?1 0 1 .. . ?1 .. .  ·Ã‚ ·Ã‚ ·  ·Ã‚ ·Ã‚ · .. . 0 0 Xn . ? ? Xn? 1 . . . 0 . . ?1 X1 1 ? ? ? ? ? ? ? ? ? Solving from the bottom up (back substitution), we have Y1 = Y2 = v 1 v X1 =? X1 = ? tY1 , ?t v 1 v (X2 ? X1 ) =? X2 = X1 + ? tY2 , etc. ?t This whole process turns out to give the same random walk sampling method. Had we not gone to the time reversed (X , etc. variables, we could have calculated the bidiagonal Choleski factor L numerically. This works for any problem with a tridiagonal energy matrix H and has a name in the cont rol theory/estimation literature that escapes me. In particular, it will allow to ? nd sample Brownian motion paths with other boundary conditions. 3. 2 The Brownian bridge construction The Brownian bridge construction is useful in the mathematical theory of Brownian motion. It also is the basis for the success of quasi Monte Carlo methods in ? nance. Suppose n is a power of 2: n = 2L . We will construct the observation path X through a sequence of L re? ements. First, notice that Xn is a univariate normal with mean zero and variance T , so we may take (with Yk,l being independent standard normals) v Xn = T Y1,1 . Given the value of Xn , the midoint observation, Xn/2 , is a univariate normal4 with mean 1 Xn and variance T /4, so we may take 2 Xn 2 v 1 T = Xn + Y2,1 . 2 2 At the ? rst level, we chose the endpoint value for X . We could draw a ? rst level path by connenting Xn to zero with a straight line. At the second level, or ? rst re? nement, we created a midpoint value. The seco nd level path could be piecewise linear, connecting 0 to X n to Xn . 4 We assign this and related claims below as exercises for the student. 8 The second re? nement level creates values for the â€Å"quarter points†. Given n X n , X n is a normal with mean 1 X n and variance 1 T . Similarly, X 34 is a 2 42 2 4 2 1 1T normal with mean 2 (X n + Xn ) and variance 4 2 . Therefore, we may take 2 Xn = 4 1 1 Xn + 22 2 T Y3,1 2 and n X 34 = 1 1 (X n + Xn ) + 2 2 2 T Y3,2 . 2 1 The level three path would be piecewise linear with breakpoints at 1 , 2 , and 3 . 4 4 Note that in each case we add a mean zero normal of the appropriate variance to the linear interpolation value. In the general step, we go from the level k ? 1 path to the level k paths by creating values for the midpoints of the level k ? 1 intervals. The level k observations are X j . The values with even j are known from the previous 2k? 1 level, so we need values for odd j . That is, we want to interpolate between the j = 2m value and the j = 2m + 2 value and add a mean zero normal of the appropriate variance: X (2m+1)n = 2k? 1 1 2 mn X 2k? 1 + X (2m+2)n 2 2k? 1 + 1 2(k? 2)/2 T Ym,k . 2 The reader should check that the vector of standard normals Y = (Y1,1 , Y2,1 , Y3,1 , Y3,2 , . . . t indeed has n = 2L components. The value of this method for quasi Monte Carlo comes from the fact that the most important values that determine the large scale structure of X are the ? rst components of Y . As we will see, the components of the Y vectors of quasi Monte Carlo have uneven quality, with the ? rst components being the best. 3. 3 Principle components The principle component eigenvalues and eigenve ctors for many types of Brownian motion are known in closed form. In many of these cases, the Fast Fourier Transform (FFT) algorithm leads to a reasonably fast sampling method. These FFT based methods are slower than random walk or Brownian bridge sampling for standard random walk, but they sometimes are the most e? cient for fractional Brownian motion. They may be better than Brownian bridge sampling with quasi Monte Carlo (I’m not sure about this). The eigenvectors of H are known5 to have components (qj,k is the k th component of eigenvector qj . ) qj,k = const  · sin(? j tk ) . 5 See e. g. Numerical Analysis by Eugene Isaacson and Herbert Keller. 9 (8) The n eigenvectors and eigenvalues then are determined by the allowed values of ? j , which, in turn, are determined throught the boundary conditions. We 2 2 can ? nd ? j in terms of ? j using the eigenvalue equation Hqj = ? j qj evaluated at any of the interior components 1 k n: 1 2 [? sin(? j (tk ? ?t)) + 2 sin(? j tk ) ? sin(? j (tk + ? t))] = ? j sin(? j tk ) . ?t Doing the math shown that the eigenvalue equation is satis? ed and that 2 ?j = 2 1 ? cos(? j ? t) . ?t (9) The eigenvalue equation also is satis? ed at k = 1 because the form (8) automatically satis? es the boundary condition qj,0 = 0. This is why we used the sine and not the cosine. Only special values ? j give qj,k that satisfy the eigenvalue equation at the right boundary point k = n. 10 How to cite Computational Efficiency of Polar, Essay examples