Isaac Rodriguez Essays - Food And Drink, Mexican Cuisine, Appetizers

Isaac Rodriguez ENC0017 May 2 4 , 2017 Professor S isto Sandra Cisneros - Reflecting on her life as an artist Sandra Cisneros begins reflecting on her as an artist beginning with a memory when she was in In France. Her friends asked her to dinner and to her surprise they were expecting her to participate in the preparation of the meal. Her job was to make tortillas, which she had no experience making. She managed to make the tortillas without her friends noticing that she had never made them before. She goes on to reflect on her memories of growing up in Chicago . Her experience traveling abroad was to her grandparent's home in Mexico with her family. She had three brothers. She credits her two older brothers with being contributing factors in her life as an artist. She attributes her upbringing to helping her become a writer. Sandra Cisneros traveled throughout Europe and the U.S. as a student and a teacher. She has had many wonderful experience. All of those experiences landed her back in Texas where she feels happiest because it reminds her of Mexico.

Free Essays on Bilingual Eduaction

For years, bilingual education has received criticism in the national media. The most recent attack has come from millionaire Ronald Unz. Unz had proposed an initiative, which would rigidly limit bilingual education in the state of Massachusetts. The implication of the initiative is that that bilingual education is a failed experiment. The proposed law would replace the current state law providing transitional education in public schools with new legislation. The new law would require (with limited exceptions) that all public school students are taught English by being taught all subjects in English, and by being placed in English-only language classrooms. The initiative will completely remove bilingual education for all limited English proficient (LEP) children in Kindergarten through fourth grade. Students older than ten may apply for a waiver from the English-only program on a limited basis. Further, the law would allow teachers to be personally sued for using a childâ€⠄¢s native language to aide in learning. If the initiative is passed, dramatic changes will clearly occur. Still, the question remains: is this initiative a good idea? This paper aims to explain the rationale underlying good bilingual education programs and to present findings about their effectiveness. There will also be discussions of criticism of bilingual programs. When schools provide children quality education in their primary language, they give them two things: knowledge and literacy. The knowledge that children get through their first language helps make the English they hear and read more comprehensible. Literacy developed in the primary language transfers to the second language. The reason seems simple: Because we learn to read by making sense of what is on the page, it is easier to learn to read in a language we understand (Smith 1994). Once we can read in one language, we can read in general. I work with ESL students recei... Free Essays on Bilingual Eduaction Free Essays on Bilingual Eduaction For years, bilingual education has received criticism in the national media. The most recent attack has come from millionaire Ronald Unz. Unz had proposed an initiative, which would rigidly limit bilingual education in the state of Massachusetts. The implication of the initiative is that that bilingual education is a failed experiment. The proposed law would replace the current state law providing transitional education in public schools with new legislation. The new law would require (with limited exceptions) that all public school students are taught English by being taught all subjects in English, and by being placed in English-only language classrooms. The initiative will completely remove bilingual education for all limited English proficient (LEP) children in Kindergarten through fourth grade. Students older than ten may apply for a waiver from the English-only program on a limited basis. Further, the law would allow teachers to be personally sued for using a childâ€⠄¢s native language to aide in learning. If the initiative is passed, dramatic changes will clearly occur. Still, the question remains: is this initiative a good idea? This paper aims to explain the rationale underlying good bilingual education programs and to present findings about their effectiveness. There will also be discussions of criticism of bilingual programs. When schools provide children quality education in their primary language, they give them two things: knowledge and literacy. The knowledge that children get through their first language helps make the English they hear and read more comprehensible. Literacy developed in the primary language transfers to the second language. The reason seems simple: Because we learn to read by making sense of what is on the page, it is easier to learn to read in a language we understand (Smith 1994). Once we can read in one language, we can read in general. I work with ESL students recei...

Economics of Transition Essay Example | Topics and Well Written Essays - 2000 words

Economics of Transition - Essay Example This competition not only transformed the contours of politics in the twentieth century but also led to the creation of two parallel and competing economic systems. Transition can be defined as the race of transformation of centrally planned economies into free markets. Such a transformation can be seeing as occurring after the political events that decimated the Soviet Iron Curtain. Transition is characterized by liberalization, macro-economic stabilization, restructuring and privatization and legal and institutional reforms. Countries have employed disparate transition models owing to the distinct initial circumstances that prevailed at the time when transition process materialized. Moreover, transition circumstances can also be seen as the product of the country’s peculiar socio-economic realities. While some countries like China embraced a slow and steady economic transformation approach, others like Russia adopted an aggressive and speedy model. It was envisaged that a sw ift approach could be undertaken for the privatization of small corporate entities as well as liberalization of prices and macro-economic stabilization, without encountering much economic problems (International Monetary Fund, 2000, p.3). The significance of transition for these economies can be attributed to the fact that any immature measures for transition tend to lead to economic collapse. This was all the more clear for example in the Russian banking sector failure in the late nineties. This paper will analyse how rent-seeking activities of financial institutions can have a bearing on hardening budget constraints in the transitional process. 2. Soft and Hard Budget constraints It was Kornai who invented the term ‘soft budget constraint’ to elucidate economic behaviour of the communist economies. This concept is deemed as the focal point of the subject of transition from socialist to capitalist economies. The term itself is derived from microeconomic terminology to shed light on a pragmatic economic social syndrome. The necessity of implementing hard budget constraints is accentuated repeatedly in relation to transition of socialist economies. The disintegration of the banking sector of East Asian economies during the nineties era can be comprehended by studying soft budget constraints (Kornai et al., 2003, p.54). The soft budget hypothesis denotes a condition under which firms incurring losses are bailed out by banks. Banks can undertake this role in both, capitalist and socialist economies. However, its repercussions in socialism make it a potential problem since banks are compelled by the state to rescue loss-making entities, for safeguarding employment or pursing other aims. The entanglement of state and banking into one entity means that any failure on the part of the banks will reflect clearly in the finances of the state. Typical failure in these circumstances leads to massive cuts in social sector development. Under a soft budget const raint, capital is injected in the loss making firm by the banks to keep them afloat and in a position to pay taxes, therefore acting as a disincentive for privatization. Research on the issue practically proves by applying a model based on the local authorities, a bank and firm (Brandt et al., 2003, p.13). The bank managers’ propensity to lend loans to private firms rather than SOEs will escalate as

Pursue litigation Essay Example | Topics and Well Written Essays - 500 words

Pursue litigation - Essay Example This ensures that people do not take advantage of legal loop holes to cause damage to others property or image. However, there may be chances where changes in the wider society happen so fast as to outpace changes in law. Tort damages are financial damages that are required from the offending party. These are required to make up for the injuries caused to the injured party. These damages differ as they are dependent on the injury and may include present or future injuries, mental distress, loss of wages amongst other tortuous behavior. There are three types of torts; intentional torts, negligence, and strict liability. Intentional torts result when a person causes damage intentionally; for instance, if one slaps another with an aim of causing harm, such is considered an intentional tort (Lunney & Oliphant, 2008). A similar act perpetrated unintentionally may not be regarded as intentional tort but might be considered a negligent act. Negligent act qualifies as tort. An important observation here is that tort does not always result from intended acts but also results from unintended acts with potentiality of harm, in this case the tort is referred to as strict liability. When a defendant is charged under strict liability, it means that their engagements, though innocent present potential liability to those around. All these classes of tort attract potential charges as they pose actual or potential injury to people. Unfortunately, not all people are aware of the existence of these and are ignorant of the likely consequences. This means that they do not know what action to take and often their rights are continually trampled. To understand the likely results and redress, such individuals require legal counsel. First, they need to understand what translates into a tort. They should seek damages for such action as these are likely to result to loss either financially or due to the distrust

Corporations and How They Have Changed From the Beginning of the Research Paper

Corporations and How They Have Changed From the Beginning of the Twentieth Century Until Now - Research Paper Example Lastly, the paper will explore when people should demand the government aid to rebuke or persuade activities of corporations. Introduction The American economy underwent significant maturity during the twentieth century. The changes in the economy were accompanied with the emergence of numerous corporations. Technocrats emerged and replaced business barons; high salaried managers became the executives in these corporations. As a result of the rising number of corporations, organized labor movement emerged. Moreover, the technological developments witnessed during the 1980s and 1990s brought with it a new culture of entrepreneurship that gave rise to tycoons (Drehle 48). With globalization, corporations had the opportunity to expand their operations to international borders. As such, companies opened foreign branches that helped to increase profitability and gain a large customer base. Discussion Through subsidies, special taxes, and special protocols, the federal government significa ntly contributes to the protection of corporations. Tax subsidies emanate from selective tax legislation aimed at benefiting some industries in the economy. The industries that reap maximum benefits from tax subsidies include corporations. Subsidies enhance the corporations to increase their returns in the market where they operate. Tax subsidies may be applied in a number of variables in a Corporation. For example, the federal government can decide to offer subsidies in terms of labor or land (Drehle 50). Special taxes and subsidies may take the form of general investment tax credits that may be made available to corporations. For example, the federal government can decide to favor productive methods that are capital-based. In its view, the federal government argues that tax subsidies play an essential role in promoting economic growth and economic stability. The federal government should make concerted efforts to develop, promote, and foster international and domestic trade by off ering subsidies to corporations (Drehle 51). As a result, trade barriers will be removed and this promotes free trade in the American economy. The reduction of trade restrictions benefits the entire economy since some trade rules hinder economic development. Tax reductions and offering of subsidies to corporations can be justified; therefore, the federal government has an obligation to subsidize corporations. Subsidies and reduction in taxes mean that the corporations will have an opportunity to manufacture goods in large quantities. Therefore, corporations will sell goods to consumers at reduced prices. As such, consumer exploitation by middlemen and businesses will be minimized. Thus, it can be concluded that distinct taxes on corporations serves to protect consumers from buying goods at extremely high prices (Drehle 51). Through the use of subsidies and tax reduction, the government guards the corporations against quiting the industries in which they operate. Some economic activi ties would not take place in the absence of subsidies and special taxes. Therefore, subsidies protect the business and normally lead to low imports and high exports (Drehle 52). The benefits of subsidies on businesses extend to employees working in these corporations. As a result of subsidies, corporations can pay their employees high wages. Consequently, the government will get high revenue from taxation of individual income. The federal gove

The Law Enforcement In England And Wales Criminology Essay

The Law Enforcement In England And Wales Criminology Essay Police officers are perceived as the gatekeepers of the criminal justice system (Mendias Kehoe, 2006:70). They are in charge in ensuring safety and security of the public, preventing crimes, and apprehending offenders. Law enforcement in the England and Wales traces its roots back to the kin policing established in the 1800s to provide protection in the communities. This form of policing flourished until increasing threats brought by terrorism compelled the government to implement changes in the police force in the 1960s. The decades following this transformation gave way to various concepts aimed at enhancing the quality of police service. Today, England and Wales follow a law enforcement style of policing in each of their decentralized forces. This paper aims to provide a critical appraisal of the policing style in the England and Wales. The next section of the paper presents the evolution of the police force. This is followed by a discussion of the different features of the England law enforcement. Critical appraisal is presented in the discussion section. Lastly, a conclusion is presented. Law enforcement in England and Wales According to Alpert and Dunham (2009), policing in England started as a private force for the elite members of society. Those who cannot afford to hire their own guards rely on a shared and informal form of policing called the kin policing. People are expected to ensure their own safety, serve as guardians of their own community, and bring forth criminal acts to the court of law. According to Beckley (2007: 938), this form of policing was employed until the 1960s when terrorism threats prompted the development of more appropriate policing strategies. The Police Act of 1964 provided for the establishment of more scientific and organized law enforcement to meet the changing needs of the public. Police professionalism standards such as the Metropolitan Police Principles and the Association of Chief Police Officers Statement of Common Purpose and Values of 1985 were established. Various innovative changes in the police force followed these developments and continued to this day. Despite these changes, policing in England and Wales still exhibit some characteristics of its older form as evidenced by the lack of centralized police system. Today, law enforcement in England and Wales consist of 43 individual county police forces. Having this system implies diversity in policing strategies between the different counties (Hawdon, 2008: 191). Moore (1992: 107) on the other hand observed that despite all this, professional law enforcement remained the main basis of any policing strategy. Features of law enforcement in UK and Wales Das and Verma (2003: 190) contend that police organizations are shaped by culture, political priorities, and immediate needs of the situations surrounding them. Law enforcement in England and Wales has distinct features that separate it from other police forces in the world. Professionalism Das Verma (2003: 204) noted that law enforcers in Europe are noted for their professionalism. They are among the best in terms of training and education. Despite this advantage, Moore (1992: 110) argued that professionalism in police force is an ambivalent concept. He believes that police professionalism should not only focus on criminal laws but also those laws that protect citizens from arbitrary action by government agencies- including the police. Thus, the police officers emphasis on the crime control rather than the law contradicts the essence of their professional duty. Multi-functioning McGarrell, Freilich, and Chermak (2007: 143-4) contend that law enforcement has changed more than any other agency in the public bureaucracy since the 1960s. The authors added that the developments in Englands police force were not only brought by threats of terrorism but also by the social and political changes in the past two decades. Police force had to adjust to meet the changing needs of the public. Police officers are not only expected to enforce laws but also to ensure public safety and security, investigate crimes, do job-related community service, control traffic, and respond to publics call for assistance (Das Verma, 2003: 224). The multitude of tasks expected from the police makes it hard to measure police performance and thus result to ambivalent perceptions about the force. Alpert and Dunham (2009) assert that people have conflicting views of the police. They are brave crime fighters and heroes on one hand and corrupt, heartless, and brutal on the other. Public perceptions about the police are often based on selective positive and negative experiences and not on their collective performance. Beckley (2007: 939) also noted the lack of centralized police organization like the National Guard or Compagnie Re ´publicaine de Se ´curite ´ in the United States and France in the UK. This function is decentralised to the various police forces. UK police officers are in charge of community policing tasks as well as specialist police duties. They have to perform tasks from traffic supervision and community patrol to criminal investigation. Technologically advanced According to Swanson, Territo and Taylor (1993), police officers not only need human and conceptual skills but also technical skills. They have to be able to use computers; analyse evidence; apply management principles; and perform planning, accounting and budgeting functions. The use of radios, mobile phones and vehicles in patrolling enables police to respond to calls for help quickly. Computers made reporting and record keeping more effective and efficient. The advancements in forensic science brought investigations to a different level. However, critics say that the use of patrol cars instead of foot patrolling limits interaction between the community and police, thus contributing to the degradation of police and community relations. Power Weisburd and Braga (2006:13) stressed the polices capability to perform their tasks without public intrusion. Despite the fact that they are established by the government and accountable to the public, they remain independent (Moore, 1992:109). Alpert and Dunham (2009) contended that police officers are provided with enormous discretionary power which may not be beneficial to the public. Das and Verma (2003:224) argued that the fact that the UN General Assembly formulated the Code of Conduct for Police Officials in 1979 attests to the authority of the police and the need to prevent abuse and misuse of this power. Reactive policing Moore (1992: 112) also recognised the reactive characteristic of policing. Police officers tend to rely on patrols and calls from the public to solve crimes. The advantage of this strategy is that it prevents the police from interfering in private lives of the people without their consent. This ensures that the participation of the police in the situation is necessary, official and acceptable to the community. However, this strategy is problematic because it implies that a crime has to be committed before action can be taken. While this is an effective law enforcement strategy, it is not a good crime control method. Moore also observed that it cannot be utilised in crimes where there are no victims or witnesses or where victims refuse to come forward. Discussion and appraisal of the law enforcement policing Based on the discussion provided above, it is apparent that the law enforcement style of policing in England and Wales has advantages and disadvantages. Although police officers in the England are known for their professionalism, their tendency to focus on crime prevention rather than law enforcement reflects misguided understanding of their professional duty. As a result, police legitimacy and effectiveness remains problematic. According to Moore (1992: 111), there is no indication that policing has delivered in their diverse functions. He noted that: there are no positive correlations between patrolling and crime prevention; police crime investigations are not always fruitful; response systems do not guarantee arrests; and criminal punishments do not always result to rehabilitation. This shows that implementation of policies like the UN General Assemblys Code of Conduct for Police Officials and the UNDP Police Act (UNDP Bureau for Crisis Prevention and Recovery, 2003) are necessary to ensure police accountability; improve efficiency and effectivity; and prevent police misuse and abuse of power. The multitude of police functions makes performance evaluation difficult and inconsistent. Beckley (2007: 939) noted the absence of specialised duty delegation in the UK police. Police officers are expected to be able to perform duties from the community patrolling to crime investigation. While this is advantageous in the overall growth of police officers, it prevents the development of specialised and quality skills that can only be learnt from constant exposure to a specific task. The multiple functions required of the police force prevent them from specializing and optimizing their potential in a given function. They are spread too thin to be efficient and effective. Technological advancements on the other hand have undoubtedly promoted the efficiency and responsiveness of the police force. They are able to provide timely response to citizen calls. However, too much reliance on technology discourages the police to depend on their own resources and hone their skills. By using vehicles instead of foot patrolling, the police loses the opportunity to mingle and connect with the people. Some police officers obtain information from the internet and database records instead of going to the field. Investigation and prosecution proceedings are relying more and more on forensic science to provide evidence. This reduces if not eliminates the need for people participation in the process. Employment of technology reduces interaction and prevents the creation of a better and more cooperative relationship between police and the people. As a result, the gap between police and community expands. People are not comfortable dealing with the police and may hesitate to call for assistance or help them in their investigations. Hillyard and Tomlinson (2000: 400) believe that police officers have political, legal and financial accountabilities. However, closer inspection of the police force reveals that police officers are only accountable to their professional mandates and not to anyone else (Moore, 1992: 116). Despite the introduction of the value for money service in law enforcement under the Thatcher government (Long, 2004), the police force still have the discretion to decide which public concerns are important and which are not part of the police work. They can refuse assistance if they deem the work to be irrelevant to police functions. This promotes public distrust to the police. Although there is increasing recognition of the need to enforce police accountability, the question about how and to whom they should be made accountable remains. The authority and autonomy given to the police force on the other hand creates dangers on the way police works. While the idea of a law enforcement emancipated from the government is alluring, the unrestrained power it afforded the police can be daunting. The police functions as an objective agency that protects the people from criminals and from the government abuses; however, there are beliefs that the people are also susceptible to dangers from the police force itself. Organisations like the UNDP recognize the authority of the police force and how this could result to abuse and misuse of power. There is a need to draw the line between accountability and authority; between policing and coercion; and between public safety and citizen liberty. Furthermore, Moore (1992: 117) also argued that the lack of governmental regulation and public control in policing also results to their stagnation. They remain oblivious of the government and citizens views, thus they are limited by their own perceptions and mandates and fail to make appropriate improvements in their services. The UK police force is now perceived as an industry rather than an agency. They have become productivity-oriented (high clearance rate, high arrest rate and high productivity) rather than community-oriented. As mentioned earlier, the lack of centralised police agency in the UK gives the police force more authority and responsibility. Their involvement is not only limited to the community but to the entire nation. They have more authority than their counterparts from other countries. They have too much power and too little regulation, thus, the risk of police domination. Reactive policing requires the police to take action after a crime is committed. This ensures that the police are not unlawfully intruding into the lives of the citizens. However, this promotes retrospective instead of preventive action. It relies on victims and witnesses in order to enforce the suitable police action. Also, reactive policing does not ensure the apprehension of the offenders. Thus, this strategy does not help reduce crime. Another issue is the deterioration of relationship between police and the public. Negative issues like police discrimination, aggression, abuse and corruption had resulted to the publics mistrust of law enforcers. This prevents the public from cooperating with police officers in community patrols and crime investigations. They do not believe that helping the police will lead to safer and more secure community. The flow of information to the police is not as strong as it used to be. According to Moore (1992:117), some people choose to stay quiet than cooperate with the police to avoid retaliation from the violators. This implies not only the peoples lack of trust on the integrity of the police force but also on their capability to protect them. Conclusion The law enforcement style of policing in the UK and Wales is both favourable and detrimental to the performance and effectivity of policing and the publics perception of the police force. It promotes professionalism of the police force. The various features of policing in England and Wales reflect the governments attempt to improve the police force and meet the changing needs of the public. The integration of various strategies such as reactive policing addresses the concern of involvement and interference. Technological advancements are incorporated for more efficient, effective, responsive, and up-to-date system. However, the multitude of functions makes performance evaluation problematic. Despite these imperfections, law enforcement style of policing in the UK and Wales is effective as it provides a common backdrop to the various police agencies in the said places.

OCeans :: essays research papers

Two-thirds of our planet is covered with oceans. The Pacific, the Atlantic, the Indian, and the Artic are the world's major oceans. They were formed by a series of geological processes that continue to affect the Earth. Seven main parts, called lithospheric plates, make up the Earth's mantle layer and crust. The plates fit together millions of years ago, and are constantly moving (at a slow pace-like a fingernail grows) over a layer of squishy, soft rock called asthenosphere that lies beneath the crust. Magma rises to fill the space when the two plates move away from each other to form an ocean. This is the way the ocean grows over millions of years. The ocean floor has an amazing landscape similar to dry land with huge mountains and deep valleys, slopes and plains, trenches and ridges. Through today's technology advances we are able to study the ocean deep and learn about it. The ocean is consistently moving. Its surface can change from calm and mirror like to wild and treacherous. Most waves at sea are caused by wind. The waves created the gales that blow during a tropical cyclone are 46ft and higher. The largest wave known to have been caused by the wind was 112ft high. Waves can also be created by volcanic eruptions. These waves are known at tsunamis. They are wide columns of water that reach down to the sea floor and can travel for great distances, at the speed of a jet plane. Colliding currents can also change the surface of the ocean. When the tide turns, the opposing currents meet and may create a whirlpool. Ocean currents are massive bodies of water that travel long distances around the world. Wind is the major force that creates currents. There are seven main ocean currents that move in large, circular streams at a walking pace. In the Northern Hemisphere currents move in a clockwise direction; in the Southern Hemisphere they are counterclockwise. Warm or cold water currents are carried by winds along the shorelines. This affects the climate of the various continents on the way. The current that carries warm water from the Caribbean Sea, up the east coast of the United Stated and then to the west coasts of Britain and Northern Europe is called the Gulf Stream. These areas would be much colder without the Gulf Stream. The "pull" of the moon and the sun also affect the ocean by causing the tides.

Computational Efficiency of Polar

Lecture Notes on Monte Carlo Methods Fall Semester, 2005 Courant Institute of Mathematical Sciences, NYU Jonathan Goodman, [email  protected] nyu. edu Chapter 2: Simple Sampling of Gaussians. 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 f unctions (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 dis tribution 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 sug gested 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 Re call 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 e xpertly 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 quasi 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. The formula X = AY w ill 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 ba ck 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 matrix 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. O ne 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 equations 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 control 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 w ith 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 second 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 eigenvectors 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 the digestive, cardiovascular and respiratory system are interrelate Essay

The human body has to be controlled from advanced systems internally of which connect in order to function effectively. There are ten main systems within the body of which all perform various activities and all interrelate with at least one other system; the nervous system. The most complex of all body systems, the nervous system controls all functions within the body. The main function of the respiratory system is to inhale oxygen and exhale the waste product carbon dioxide. Oxygen is breathed through the mouth and nostrils into the lungs. The gas then diffuses through the alveolar walls and into the red blood cells. Carbon dioxide carried back within erythrocytes diffuses back through the alveolar walls and is exhaled out through the mouth. The cardiovascular system has a role of circulating blood around the body. The contents of blood being circulated include nutrients, gases and waste products. The main organs involved within this process are blood, arteries, veins, heart and the lungs. The function of the digestive system is to absorb and adapt key nutrients which are required to maintain and regulate other systems in the body. Enzymes secreted within the digestive system have a role of breaking down food from proteins into amino acids, starch into glucose and fats into a mixture of fatty acids and glycerol. It can be analysed that the two body systems work together. Which are the respiratory system and the cardiovascular system. Oxygen is inhaled and enters the respiratory system. It travels down into the lungs through the trachea. Once inside the lungs, the gas enters the alveoli and diffuses through the alveolar wall. After it diffuses through the capillary wall, it enters the cardiovascular system. Oxygen then binds with haemoglobin forming oxyhaemoglobin and is then transported around to cells within the body. Carbon dioxide is transported as a waste product within red blood cells through the cardiovascular system. This needs to be exhaled as too much carbon dioxide present within the blood causes acidosis. The gas must vacate  from the blood through the same route oxygen entered the blood. It must exit through the respiratory system. Carbon dioxide diffuses back into the alveoli and then is exhaled out of the mouth through the respiratory system. It is evident that both systems are required in order to sufficiently exchange gas. If the human body was unable to perform gas exchange, cells within the body would die so this interrelation is important. The respiratory system interrelates with the digestive system to perform a defence mechanism. When a micro-organism has successfully entered the human body, goblet cells along the respiratory tract have the ability to produce a larger yield of mucous. Mucous is a defensive â€Å"sticky† liquid of which has the ability to collect micro-organisms that bind with the substance. Cilia within the trachea slide the mucous towards the entry of the oesophagus. The substance is then delivered through the tube and into the strong hydrochloric acid present in the stomach. To aid with the respiratory system, the digestive system provides the diaphragm with key nutrients necessary to facilitate breathing. It is also noted that both the digestive system and respiratory system provide the key products to perform aerobic respiration. The cardiovascular system interrelates with the digestive system. Various nutrients produced from the digestive system are required to maintain the heart rate within the cardiovascular system at a normal rhythm. The nutrients are essential within the whole body so once they have been churned; they are absorbed within the blood and then transported via the cardiovascular system to their required destination. When the digestive system requires more blood, vessels expand to acquire the demands. The circulatory system carries chemical signals which control the speed of digestion. Toxins produced by the digestive system are transported via the cardiovascular system to the kidneys to excrete the material. It is identifiable from analysing the roles of the systems that both are required in order for the human body to remain in a healthy state. Also they’re all linked by being part of one and the same organism. Respiratory system – Oxygen is inhaled, cardiovascular system – oxygen is carried to every cell in the organism, nutritive substances are carried from the digestive system to every cell in the organism. Cardiovascular system – carbon dioxide is  carried to the respiratory system and it is exhaled. Cardiovascular system – whatever remains of the nutritive substances after they’ve been processed in the cells is returned to the digestive system and is â€Å"thrown away† at the end of the digestive system.

Law Day History, Origins and Celebration Traditions

Law Day History, Origins and Celebration Traditions History of Law Day In the USA, May 1 is marked on the calendar as Law Day. It is a national day when the country celebrates the role of law and its contribution to the foundation of Americans freedoms. May 1 was primarily chosen as an international day, known as May Day, to commemorate the attempts of workers to improve their working conditions. Though, with the span of the time many European countries continued to stick to the traditions and carry out celebration on May 1, while some other countries, including the USA, started to observe significant to their nation Labor Day on the first Monday of September. The idea to celebrate Law Day belongs to Charles S. Rhyne, the former president of the American Bar Association. He was a legal counsel of President Eisenhower in 1957-1958. Rhyne offered to celebrate Law Day to recognize the US legal system. Since Law Day was proclaimed as a national day by President Eisenhower in 1958, it was stated that henceforth May 1 would be the date to observe it. Law Day has become a reminder for everyone to appreciate the virtues the American nation has, the equality and justice under the law in the relations with each other and other countries and to cultivate the respect for law necessary for the maintenance of the democratic society. Law Day Festivals and Celebrations , the theme will be The 14th Amendment: Transforming American Democracy. There is also a tradition, which was established by President Eisenhower in 1958 and is observed by every President till present time: to issue an annual Law Day Proclamation. Each Proclamation is aimed at discussing the American rights and American law and stresses the importance of the recognition of the rule of law as supreme as in Eisenhowers Proclamation; calls for struggle to defend the way of life in Kennedys Proclamation; aspires to fuller awareness of the importance of respect for law in Johnsons Proclamation; advocates to make the country a beacon of liberty in Reagans Proclamation; celebrates the nations commitment to the rule of law and the equality under the law in Obamas Proclamation. Broaden your outlook and learn about the origin and celebrations of our countrys national holidays!

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