Commentary on Women Beware Women Essay Example for Free

Commentary on Women Beware Women Essay Beware Women is a Jacobean tragedy, which has a complex plot and deals with corrupted characters. This tragedy is about corruption in the court and life in general, love by money and how women can lead other women to destruction. As Tricomi states about the characters, they ‘are not wholly the product of their circumstances, but their circumstances condition their choices and propel them toward their destiny’. Middleton, as other Jacobean playwrights before him, has managed to point out that ‘aristocratic life is brutal and corrupt’. 2] The focus of this commentary will be concentrated on marriage and to what can lead one to fall apart. The first scene (Act I), deals with the characters of the play’s main plot: Leantio, Bianca and the Mother. In this scene, the readers come across with the insulting behaviour towards Bianca, where she is treated as an object. Leantio speaks of his wife with words of business, to him she is â€Å"the most unvalued’st purchase†.He describes her as if she was a dangerous object that must stay hidden and safe, away from the sight of men. When he talks about her, it is obvious that Bianca is for him a treasure and he is the thief that now has to hide his â€Å"best piece of theft† (I. ii) in a safe place so no one will steal it from him. Such words describing a human being are rather cruel, especially when Leantio is talking about a person for whom he is supposed to have true and pure feelings of love. Leantio is aware that Bianca’s family is rich, but he also knows that by marrying Bianca in secrecy and taking her away from them, Bianca will lose all of the property and money that belongs to her. He has also written over to her his house and put his mother in jeopardy. Although his act seems a romantic one and, even though he speaks of that relationship and feelings as being pure, his love is not mature; rather, it is one filled with jealousy. In the beginning of the play Bianca could be characterised as the victim because she has a mother-in-law who is not fond of her and does not approve their marriage and she is now imprisoned in poverty and in home. However, Bianca is ‘as much a victim as perpetrator, and she is to be judged as a tragic protagonist with a vexing mix of virtues and flaws’. [3] As seen in the plot, the Mother aids and abets in Bianca’s meeting with the Duke. The Mother and Livia hatched up a plan for Bianca’s rape and she falls into the trap, as Isabella did, but the rape was almost enticed on her part. ‘The attitude towards Bianca is one of dehumanizing possession and manipulation’. [4] But Bianca, after that, changes drastically and soon enough she becomes one of the most corrupted characters, who along with others, brings about the downfall and the final bloodshed in the play. Bianca chose money over her marriage, although, she blames the other women for her disaster. ‘Treachery and betrayal [ ] are Bianca’s terms of explanation for her downfall’. [5] Bianca is seen by her mother-in-law as an added burden to her son’s finances. The Mother’s interest is focused only on money. For the Mother, Bianca as a wife has nothing to offer, she will only demand and receive. The Mother is sizing up the economics of their situation now that there are three members in the family. Leantio can barely support himself, and up until now he had to support his mother as well. The Mother doubts that her son is able to support a family of three. She claims that nothing can save him from this financial dead end by saying â€Å"My life can give you But little helps, and my death lesser hopes† (I. i). The Mother thinks of Bianca, as for every other wife, that she will require from Leantio â€Å"maintenance† (I. ii) fitting to her â€Å"birth and virtues† (I. ii), but also gratification of her desire for â€Å"affections, wills, and humours† (I. ii). [6] Leantio then expresses his intentions towards Bianca, by replying to his mother’s words, pleading with her not to â€Å"teach her to rebel† (I. ii) now that â€Å"she’s in a good way to obedience† (I. ii). Leantio’s â€Å"assurance† (I. ii), of keeping his â€Å"jewel† (I. ii) locked away â€Å"from all men’s eyes† (I. ii), is his mother. She is the one who holds the â€Å"key† (I. i) to his â€Å"treasure† (I. ii), and â€Å"old mothers† (I. ii) are â€Å"good to look to keys† (I. ii) when â€Å"sons lock chests† (I. ii). However, the irony here is that later on, it’s the Mother herself who pushes Bianca towards rape with the Duke, first to get rid of her, but then to accrue some of the benefits from the court life for herself and her son. Bianca is to Leantio nothing but an object of â€Å"great value† (I. ii), a â€Å"matchless jewel† (I. ii) that he has stolen. Because â€Å"temptation is a devil will not stick to fasten upon a saint† (I. ii), Leantio’s â€Å"gem† (I. ii) must stay hidden and locked. This is the â€Å"great policy† (I. i) for Leantio in order to never lose a treasure; never â€Å"show thieves our wealth† (I. ii). Bianca is the â€Å"treasure† (I. ii), Leantio is the â€Å"thief† (I. ii), and the â€Å"key† (I. ii) to his happiness holds his mother, thus, it could be said that the chest with the key is symbolic of Leantio and Bianca’s relationship. To conclude, this tragedy proves that women should beware women. Women lead other women to destruction, and are even responsible for another woman’s rape. Corruption and enemies are present everywhere but, as Bianca says in her dying breath, â€Å"Like our own sex, we have no enemy†.

Tourism in Sri Lanka

Tourism in Sri Lanka 1. Sri Lanka is an island country and god has given to Sri Lanka blessings of nature blue seas, green hills, sandy beaches, abundant wildlife, cascading waterfalls, a riot of flowers and fruits, coconut groves, tea estates, spice gardens. Here ones day can be gloriously lazy by a palm cool lagoon or full of joy, swimming, snorkeling, and fishing or sailing like a brilliant jewel in the sparkling waters of the Indian Ocean. Sri Lanka is located in a strategically considered naval spot on the world map. This significant situation creates a path for a major sea route. In addition it creates a strategic naval link between West Asia, Africa and East Asia. Sri Lanka entered the international tourism arena in the 1960s. Since then, government involvement has been the key factor in tourism development in Sri Lanka. The Ceylon Tourist Board (CTB) was established in 1966 in order to provide direction and leadership to this promising sector of the developing economy. The country is famous for h ealthy and cultural heritage. Hospitality, tropical forests, natural and fascinating beaches, archeological sites and the tropical climate are the more attractive features. These factors make Sri Lanka a wonderful tourist destination. Over the years tourism in Sri Lanka has developed significantly. Today tourism has become the sixth Foreign Exchange Earner (FEE) in Sri Lankan economy. 2. Tourism can make a great impact on development of the Sri Lankan economy. Tourism is a fast growing industry which has been identified presently. Employment opportunities and regional development are playing significant role with regard to the tourism industry development. It can be considered as the most important industry in the country as a whole. The governments vision is to make the tourism sector as Sri Lankas most innovative and profitable income generating source. 3. During past three decades Sri Lankas tourism industry had few draw backs. This was mainly due to the security situation which prevailed in the country. Further the tourism industry was also affected due to the Tsunami catastrophe which occurred in 2004. Approximately 2/3rd of the coastal area including tourist infrastructure facilities was destroyed. The immediate post war period gave rise to a dramatic increase of tourist arrival from end May 2009. 4. In order to support the efforts and to develop the tourism sector in Sri Lanka there should be hotels with exclusive facilities which are in par with the international standards to accommodate a large number of visitors. In the year 2010 approximately one million tourists visited Sri Lanka. Unlike in the past, the inflow of tourists is increasing day by day as a result of the end of the 30 year civil war which destroys the whole country. Following graph indicates how tourism increased from 1966 to 2011 in Sri Lanka. The government is expected to receive 2.5 million tourists in 2016. To cater for this increase tourist hotels in Sri Lanka must be developed. Similarly other facilities should also be increased to meet the desired economic goals. AIM 5. The aim of this paper is to analyze the development of Sri Lanka tourism -challenges and opportunities GEOGRAPHY AND CLIMATE OF SRI LANKA 6. Sir Lanka has a various terrain but it mainly consists of flat lands but south-central portion of the countrys step sided river, canyons interior features and mountain. The flatter regions are the areas where most of Sri Lankas agriculture takes place, aside from coconut farms along the coast. Sri Lankas climate is tropical and the southwestern part of the island is the wettest. . The northeastern part of Sri Lanka is drier and most of its rain falls from December to February. Most of the rain in the southwest falls from April to June and October to November Sri Lankas average yearly temperature is around 86 °F to 91 °F. An important geographic note about Sri Lanka is its position in the Indian Ocean, which made it vulnerable to one of the worlds largest natural disasters. On December, 26, 2004, it was struck by large tsunami that hit 12 Asian countries. Around 38,000 people in Sri Lanka were killed during this incident and much of Sri Lankas coast was destroyed. 7. Sri Lanka is situated in strategic location near major Indian Ocean sea lanes. It has a total area of 65,610 km ², with 64,740 km ² of land and 870 km ² of water. Its coastline is 1,340 km long. Sri Lankas climate includes tropical monsoons: the southwest monsoon (June to October) and the northeast monsoon (December to March), its terrain is mostly low, flat to undulating plain, with mountains in the south-central interior. The highest point is Pidurutalagala at 2,524.13 m. Resource include limestone, mineral sands, gems, graphite, phosphates, clay, and hydropower. 8. Adams Bridge, a land connection to the Indian mainland, is now mostly underwater with only a chain of limestone shoals remaining above sea level. According to temple records, this natural causeway was previously complete, but was breached by a violent storm (probably a cyclone) in 1480. SRI LANKAS COMMITMENT TO TOURISM 9. Tourism is a vital area in the policy structure has been identified as capable of efficiently driving the countrys socio economic development. The programmes vision for the tourism sector is to make Sri Lanka foremost freedom destination in the South Asian Region. The programme believes the human resources and natural and cultural endowments values and nation will be essential in transforming Sri Lanka into a centre of excellence and offer tourists the highest values of real experiences in its unique setting. 10 The One Stop Unit Unit for National Investment in Tourism is a Centralized promotion and facilitation center Established Thurs assist potential tourism investors interested in investing in Sri Lanka Tourism Industry. Specialist staff from various government agencies helps investors Identify possible projects, Obtain information possible, Regarding Investments, submit applications, and Provide support in obtaining investment promotion privileges, trade licenses and other approvals Requested for project clearance. OSU Prevents the need for investors Thurs spend time in search of answers and ensures all queries are handled by its specialist staff. 11. Serious civil disturbances starting in July 1983 and the subsequent violence badly affected tourism. Total arrivals were 230,106 in 1986, down 43 percent from 1982. To ease the dilemma of the industry, the government provided various concessions to hotels, such as the rescheduling of loans and the reduction of the turnover tax from 10 percent to 5 percent. The Ceylon Tourist Board also undertook a crash promotion program in an attempt to restore the islands image in world tourist markets. Tourist arrivals in the first six months of 1987, however, showed a decline of 23 percent compared with the same period the previous year. In early 1988, the outlook was for further contraction. 12. In 1988 it remained unclear whether the policies of economic liberalization Sri Lanka has pursued since 1977 would succeed in their principal goals of employment, wealth creation, and economic diversification. Although increased rice production, the growth of textile manufacturing, and an improved infrastructure were successes that could be attributed to the post-1977 policies, these gains came at the cost of a mounting foreign and domestic debt and declining living standards for the poor. POLITICAL ENVIRONMENT TOWARDS TOURISM 13. The One Stop Unit Unit for National Investment in Tourism is a Centralized promotion and facilitation center Established Thurs assist potential tourism investors interested in investing in Sri Lanka Tourism Industry. Specialist staff from various government agencies helps investors Identify possible projects, Obtain information possible, Regarding Investments, submit applications, and Provide support in obtaining investment promotion privileges, trade licenses and other approvals Requested for project clearance. OSU Prevents the need for investors Thurs spend time in search of answers and ensures all queries are handled by its specialist staff. 14. Serious civil disturbances starting in July 1983 and the subsequent violence badly affected tourism. Total arrivals were 230,106 in 1986, down 43 percent from 1982. To ease the dilemma of the industry, the government provided various concessions to hotels, such as the rescheduling of loans and the reduction of the turnover tax from 10 percent to 5 percent. The Ceylon Tourist Board also undertook a crash promotion program in an attempt to restore the islands image in world tourist markets. Tourist arrivals in the first six months of 1987, however, showed a decline of 23 percent compared with the same period the previous year. In early 1988, the outlook was for further contraction. 15. In 1988 it remained unclear whether the policies of economic liberalization Sri Lanka has pursued since 1977 would succeed in their principal goals of employment, wealth creation, and economic diversification. Although increased rice production, the growth of textile manufacturing, and an improved infrastructure were successes that could be attributed to the post-1977 policies, these gains came at the cost of a mounting foreign and domestic debt and declining living standards for the poor. PROMOTION OF TOURISM INVESTMENT 16. The Board of Investment has introduced incentives in the form of tax exemptions, duty-free imports and the relaxation of controls on foreign exchange holdings. The incentives provided by the government have mainly attracted investment in the hotel sector. The present hotel capacity is 13,670 rooms. That figure is projected to increase to 21,000 rooms by 2004, which will accommodate the target of 1 million tourists. Five regional domestic airports to be restructured at a total cost of approx. Rs. 2bn: Rathmalana, Koggala ,Ampara , Trincomalee Jaffna (Rathmalana will be developed as a City Airport) Previous year. This denotes that tourism industry is one of the core sources of foreign exchange earner in economy of Sri Lanka. Due to the development of tourism improved accessibility via new highways and conversion of military to domestic airports may make high-end hotels in the Deep South and the East of Sri Lanka viable alternatives to the Galle area. OPPORTUNITIES FOR TOURISM INDUSTRY 17. Tourism directly affected to economy of a country. The momentum of growth in tourism, as result of the post conflict peaceful environment, the tourist arrivals grew to 654,476 in 2010 surpassing the previous record of 566,202 in 2004. Tourist nights one of an important indicator in measuring the volume of tourist traffic, has recorded 6, 544, 760 in 2010. It shows 60.6 per cent increase compared to 2009. Total receipts in 2010 amounted to US $ 575.9 million as against US $ 349.3 million recorded for the year 2009. It indicates a vast increase of 64.9 per cent than THREATS FOR TOURISM INDUSTRY 18. Development of tourist hotels in Sri Lanka will extremely support the rapid economic development. Developments of tourist hotels, increase room capacities, increase occupancy rates and develop the condition of rooms and upgrade the service provided in the hotels are the key factors to be mainly focused on if to develop tourism. It indeed contributes to the development of tourism and in turn development of economy of the country. Tourist hotel industry should be developed in most of the famous city for tourism with sufficient amount of tourist hotels 19. As political risk introduces additional elements of uncertainty into the rules governing tourism investment projects, the risk of capital loss is raised for longer- term projects. Political risk also negatively influences the timing and pricing of the tourism production process. Negative images, lack of foreign exchange for tourism development, lack of skilled manpower, weak institutional frameworks for tourism planning, political instability caused by communal violence, civil war conflicts are inhibitors to tourism development. However, little is known about how international tourism firms perceive political risks and other general barriers and threats to tourism promotion in Sri Lanka. 20. Common political factors which affect the tourism industry are revolution, civil war, factional conflict, ethnic violence, religious turmoil, widespread riots, terrorism, nationwide strikes, protests, cross- national guerrilla warfare, world public opinion, repatriation restrictions, bureaucratic politics, leadership struggle, high inflation, border conflicts, high external debt service ratio and creeping nationalization. Last thirty years Sri Lanka also faced the civil war and it affected the tourism industry of Sri Lanka. Organizing elections frequently also create threat to the tourism industry because during the election period in some places to control the situation cur hews are implemented. 21. This will create negative image in the mind of tourists. Some of the foreign countries have reject Sri Lankas products in their country due to the violation of human rights during the war time. For example government invested and spent on IIFA programme but it was not a successful event for Sri Lanka because the big stars have boycotted the event. They didnt visit Sri Lanka because of the protests held in their country by asking them not to visit the place. Currently Sri Lanka is having negative relationship with some foreign countries therefore the export and import of Sri Lankan products will be decreased and it affects the tourism industry as well. FUTURE TRENDS, CHALLENGES 22. Source markets in Asia and Middle East to drive demand: Sri Lankas proximity to source markets such as India and its connectivity to the Middle East and China will help in sustaining tourism growth. The rise in per capita income and therefore consumer spending in these source markets will aid the growth in tourism. 23. Infrastructure growth to fuel tourism growth: To ensure rapid growth of tourism in the future, tourism projects will need to be balance with infrastructure development. We expect that with improvements in road infrastructure and development of the new airport in Hambantota, various new destinations will emerge in the country, especially to the north and the east, as these areas have abundant natural beauty to attract tourists but currently suffer from poor connectivity. We expect more airlines to operate in the country as the second airport develops. 24. Greater competition will facilitate growth: We also expect greater competition with the introduction of new hotels in popular destinations such as Colombo, Kandy, Bentota, and Sigiriya, and Galle. We expect the existing hotels to undertake phased renovations to effectively compete with the upcoming hotels. Also, with the increase in income from hotels over the past few years, we expect an increase in the number of hospitality related transactions and mergers and acquisitions. 25. Challenge of attrition to other destinations: Currently, the hotel industry witnesss significant employee attrition to countries in the Middle East and to the Maldives and India. However, with a more stable political environment and improvement in quality of living in Sri Lanka, we expect the attrition to gradually 26. Shortage of skilled labor: As new players enter the market, the projected growth in tourism will be accompanied by shortage of skilled labour. We anticipate the need for relive hospitality education institutions as the contribution of the tourism sector to the overall economy increases. SUMMARY 27. Sri Lanka is located in a strategically considered naval spot on the world map. Sri Lanka entered the international tourism arena in the 1960s. Since then, government involvement has been the key factor in tourism development in Sri Lanka. Over the years tourism in Sri Lanka has developed significantly. Today tourism has become the sixth Foreign Exchange Earner (FEE) in Sri Lankan economy.Tourism can make a great impact on development of the Sri Lankan economy. Employment opportunities and regional development are playing significant role with regard to the tourism industry development. The governments vision is to make the tourism sector as Sri Lankas most innovative and profitable income generating source. 28. During past three decades Sri Lankas tourism industry had few draw backs. In the year 2010 approximately one million tourists visited Sri Lanka. Following graph indicates how tourism increased from 1966 to 2011 in Sri Lanka. To cater for this increase tourist hotels in Sri Lanka must be developed. Sri Lankas climate is tropical and the southwestern part of the island is the wettest. . Around 38,000 people in Sri Lanka were killed during this incident and much of Sri Lankas coast was destroyed. 29. Total arrivals were 230,106 in 1986, down 43 percent from 1982. The Ceylon Tourist Board also undertook a crash promotion program in an attempt to restore the islands image in world tourist markets. This denotes that tourism industry is one of the core sources of foreign exchange earner in economy of Sri Lanka. Tourism directly affected to economy of a country. Development of tourist hotels in Sri Lanka will extremely support the rapid economic development. Developments of tourist hotels, increase room capacities, increase occupancy rates and develop the condition of rooms and upgrade the service provided in the hotels are the key factors to be mainly focused on if to develop tourism. Tourist hotel industry should be developed in most of the famous city for tourism with sufficient amount of tourist hotels 30. Negative images, lack of foreign exchange for tourism development, lack of skilled manpower, weak institutional frameworks for tourism planning, political instability caused by communal violence, civil war conflicts are inhibitors to tourism development. However, little is known about how international tourism firms perceive political risks and other general barriers and threats to tourism promotion in Sri Lanka. 31. Last thirty years Sri Lanka also faced the civil war and it affected the tourism industry of country. Some of the foreign countries have reject Sri Lankas products in their country due to the violation of human rights during the war time. Currently Sri Lanka is having negative relationship with some foreign countries therefore the export and import of Sri Lankan products will be decreased and it affects the tourism industry as well. 32. Sri Lankas proximity to source markets such as India and its connectivity to the Middle East and China will help in sustaining tourism growth. To ensure rapid growth of tourism in the future, tourism projects will need to be balance with infrastructure development. We expect the existing hotels to undertake phased renovations to effectively compete with the upcoming hotels. We also expect the hotel companies, especially the domestic companies located in the country to undertake long and medium term hospitality training programs and regularly undertake competition benchmarking for compensation to retain the talent. RECCOMONDATIONS 33. These are the recommendations which we can produce in related to tourism. Develop a marketing strategy: Emphasizing the creation and sustaining of a positive image of Sri Lanka as a substantive and well-established tourist destination offering facilities experiences that are distinct Sri Lankan products. Establish a domestic tourism strategy: To expand the existing product base and encourage a greater level of holiday movements in addition to pilgrim movements. Develop goods and services: To meet the increasing needs of the tourism industry, and thus optimize the economic benefit of tourism development to Sri Lanka. Modify the organization of tourism activities: In order to bring about related improvements in the private sector, increase coordination between tourism and interrelated sectors, and introduce streamlining to facilitate the development of the tourism sector. Emphasize the environmental, social and cultural aspects: Tourism Development Planning is the only way to solve many issues related to tourism in development. Therefore the tourism planning process should be designed to produce goals and objectives for the destination area related to tourism development. The tourism development plan provides overall guidelines for development and identifies development opportunities. Many development countries are involved in tourism planning. Financial Assistance: Financial assistance for hotel owners and any other profession in tourist sector would help to create more employment opportunities. Domestic Airport: Airport also more than 100 km away from the tourist destination it is needed have domestic airports. Surfing Training: This will create more employment opportunities and generate good income. Among the tourists there are some people they love for surfing and keen to learn. Conduct Language Teaching: Language barrier should be overcome by conducting specially job oriented language training. Employ Combat Divers: Surfing is risky and adventure sport it dont take much time for some to happen therefore as precautionary measure it is needed some combat divers close to the surfing area it would enhance the confidence of people who do surfing. Proper Taxi Service: Taxi service should be reformed introducing good vehicles fixed with meter system and giving training for drivers. Improve Infrastructure Facilities: Infrastructure facilities such as Power, Highways, Railways, Airports, Ports, Water Supply, Telecommunications, Accommodations, and Recreational, Shopping and Banking to be improved. Awareness Programs: Tourists should be informed on specific social, cultural and religious environment programmed. GAD ALWIS SLE Maj STUDENT OFFICER -DSCSC

A Curse and a Gift :: Personal Narrative Death Papers

A Curse and a Gift He stands in the dark, lingering. Always linger. Faceless, but his cold, menacing eyes illuminate from the depth of the abyss. He’s gone now. My attention is diverted to a lonely, deserted road. There’s no sign of life anywhere. It’s pitch dark and I’m walking. I don’t know where I’m going, but I have to get out of this darkness. A cold, unsettling breeze emerges from a warm, summer’s night. I feel chills rising through my spine. â€Å"Something’s wrong.† I walk for a considerable amount of time. Finally, out of nowhere I see a speck of light. I get closer. A somber crowd stands in a circle with their heads down, as if at a funeral. Two cars, a red Pontiac Firebird, and a silver Toyota Celica, had lost control and gotten into a terrible collision. They look familiar. I force my way through the spellbound, immobile crowd. My heart beat fastens to a thousand beats per minute. I drop to my knees in hysteria. â€Å"Kelvin! Kelvin! Not you too! Please don’t do this to me!† He lays there. Face up towards a glorious sky. There are millions of stars out tonight. But I haven’t noticed them. He just lays there. His hands folded and resting on his thigh, he lays in a pool of blood, on the dry, brown earth. He hasn’t suffered any substantial physical injuries from what I can see. He looks peaceful, so peaceful, in eternal repose. I can still recognize him. He’s 6’1, with a dark, chocolate complexion, an athletic build and a handsome face. He is my brother. Claudia, a close friend of mine, comes up to me in tears. She puts her arms around me and says â€Å"I’m so sorry.† She and Kelvin, both proud thrill-seekers were racing. She has survived the collision. My brother is gone. I woke up sweating and distraught, in tears. My heart was still racing. I frantically recited psalm 23, the Lord’s Prayer. After that, I grabbed the television remote which I had placed under my collection of stuff animals and turned on the television to Cartoon Network. Tom and Jerry was being showed. I reached for my journal which lay next to the remote on my bed, and began to write. Imagine losing your uncle, cousin and a very close family friend in one year. Better yet, imagine losing four people that are very dear to you in a three year time span.

Barriers Faced By Pakistani SMEs in Raising Bank Finance :: Business, Banks

1. Introduction The SMEs globally, are recognized as engines of economic growth and play a pivotal role in boosting the economy. The importance of the SMEs sector is well recognized and its Contribution is relevant in achieving several socio-economic objectives, such as employment generation, contribution to national output and exports, and fostering new entrepreneurship. SMEs contribute in economic growth of both developed and developing countries, as they: Provide low cost employment since the unit cost of persons employed is lower for SMEs than for large-size units (Sadaquat and Sheikh, 2010). The SMEs sector’s growth of output trend to decrease in recent years since liberalization and adjustment policies (Bari and Haque, 2008). So, it is important to specially address the policy issues regarding lending toward the SME sector. This sector is facing severe problem in financing, regulatory aspects, access to non-financial inputs. Some recent trend shows that Government policies have discriminated against small-scale enterprises (Raza and Murad, 2010). There is nothing wrong with a situation in which inexperienced entrepreneurs are unable to get institutional credit. In the same study he shows that, the relative decline of small-scale enterprises in most developing countries has been accelerated by the industrialization policies adopted in these countries (Bari and Haque, 2008). Protection, regulatory constraint, investment incentives, credit control, and the promotion of industry in the public sector have all discriminated against the small. Especially, facilities reg arding small groups like female were poor and create adverse impact on the growth of SMEs (Sadaquat and Sheikh, 2010). The common idea that the cost of capital is very high for small enterprises is overly simple (Basu, 1998). A research of World Bank suggests the existence of financial constraint because formal banks do not lend to the smallest firms in most countries. It has also severe impact on the smallest firms. Access to equity and formal debt financing has repeatedly been identified as a recurring constraint to SME growth and development. Commercial banks apply conservative policies in lending to SME. More, importantly the existing structure of financial sector was developed to serve medium to large enterprises which are organized as a formal business (Kon and Storey, 2003). Most banks prefer to hold risk free-income generating assets and lending to SME is unattractive due to a range of objective and subjective factors. These include high transaction costs, inability to do away with tangible collateral requirement, no linkage of financial products with sector needs and the inability to structure/ offer and manage risk-prone SME specific medium to long term financing options.

Project Proposal Essay examples -- GCSE Business Marketing Coursework

Project Proposal INTRODUCTION Topic and Purpose This proposal identifies the need for a document, which will propose and develop reasons why a for-profit transportation service for Kent should be perused. This service should be considered because of the recent parking and commuting problems both on and off campus. There have been many reasons why on-campus parking has become difficult in recent years. Increased student enrollment at Kent State University has posed many problems. The parking lots cannot facilitate the growing need for vehicle parking. Campus officials have undertaken construction projects to curb the parking problem, but these renovations have closed lots in the near term. Also, the increased enrollment of students has caused overcrowding on campus buses. These on-campus parking problems also have indirectly affected off-campus parking problems. Off-campus parking in the city of Kent can be described as dismal. The bus service provided by Kent State University is not geared to individual parking needs and lacks the spontaneity students often crave. Increased challenges of off-campus parking include a higher degree of regulation from City Hall. Furthermore, limitations of guest parking at apartment complexes and leased property have increased at an increasing rate over the past three years. These off-campus and on-campus parking problems serve as catalysts for commuting problems across the board. The transportation problems of Ken...

Foreign Language Instruction Should Begin In Kindergarten Essay

For years it was thought that learning foreign language is getting more and more important because of the globalization. In order to acquire a new language as soon as possible, amounts of parents would like to let their children begin to learn new languages in kindergarten. There are many positive effects if people learn foreign language when they are young children. Even though adults have advantages in learning new languages as well, children learning language easier than adults is still a common thing. This essay examines firstly that obtaining a new language benefits children increase their critical thinking skills and children are good at imitating native pronunciation then considers the advantage about adults learning additional language followed by the drawback of children learning new language in kindergarten. First of all, learning foreign language in kindergarten benefits young children to enhance their brains. Catherine Ford mentioned that the brain can been seen as muscle, the more exercises it does the better it functions, (2014). Every part can be trained even human brains. Take athletes as an example, every athletes need to do the exercises to build up their muscles, seems like the more exercises they do the stronger they are. However, they begin their sport area probably when they are teenagers. One thing should be known that human brains is the first part begin to strengthen, so it is a strategic decision to begin foreign language instruction in kindergarten. Moreover, young children do not need to learn difficult words in kindergarten, which makes these children are more interested in new language without any pressure. Basically children are always interested in fresh things, so they would like use what they have just learnt, which helps them to memorize new language words. It is clear to see that children could obtain a new language when they play soma games in kindergarten with their  peers. Some pupils are rebellious when they grow up, so it is a good chance to teach them language when they are in preschool. In addition, young children are really good at imitating native pronunciation, which is also an extremely crucial reason. ‘Children who learn a language before the onset of adolescence are much more likely to have native-like pronunciation’ (Baker, F n.d.). Compare to the adults who learn foreign language, it is well known that adults have experienced how to learn a language, but they always use the origin language pronunciation to learn foreign language. There are different grammars in different languages as well, so using the same way to learn new language wastes time sometimes. Another thing is that adults cannot only learn some simple words, because when they have a conversation with someone else they need to know many vocabularies of different aspects. All of these words’ pronunciations need to be memorized, which is a tough question for adults. Although children have to learn more words then adults when they grow up, they can still deal with it because they have already learnt the foundation of the language and made steady progress step by step. Perhaps it can be seen that children learn new language more easily than adolescence. On the other side, it should be doubted that the ability of adults to obtain a new language is weaker than the one of children. Basing on the word from Mo Costandi ‘The brain’s neuroplasticity decreases with age, but this shouldn’t put off older learners.’ (2014). Although the brain of an adult is different than the brain of a child, which is totally developed and might become more mature and more ineradicable toward aged issues even more resistant to a new thing. Human are never too old to learn, the ability of brain is immeasurable, scientists believes that the brain is a highly dynamic structure, which changes itself in response to new experiences, and adapts to injuries. (2014) Comparing to children, with large lives experience, adults can learn a new word by searching a relevant memory, that is the reason why learning a new language’s vocabulary is much easier for adults than learning the grammar and syntax. While without enough experience, children are difficult to understand a new word. Moreover, learning a new language is good for the health of the brain, some research suggested that pick up a second language can decreased the change to get a senile dementia for older learner. Secondly, children learn foreign language could forget their first language. It means if young children learn second language and practice with their partners for a long time, the mother language could be forgotten. ‘Research has shown that it is important to develop a child’s first language literacy before introducing a second language’ (APEC 2006). In this aspect, adults do it better than preadolescence, because they have used their mother languages for several decades. This research shows that human can have a deep memory with the languages what have been used for many years. By the way, if foreign languages are provided in kindergarten or preschool, most of children probably are going to lose their own mother languages. Sometimes languages could be seen as cultures. Part of parents would like to see their children studying foreign language in kindergarten only because they want to make their children have more advantages than children’s competitors in the future. However, these children’s foreign languages could not better than native speaker’s, and they lose their countries’ cultures. All of these illustrate children begin to learn foreign language in kindergarten is not suitable. It has been seen that, learning foreign language in kindergarten brings children not only the increase of critical thinking, but also the development of imitating skills. However, younger children could get more pressure when study extra language course and adults have experienced how to learn a language. Statistics show that there is still an increase in children learning a new language. More research needs to be carried out on implications of this trend for children learning foreign language with making their minds more creativity and flexibility. Reference list APEC 2006, Language learning and age, APEC, viewed 24 February 2015, http://hrd.apec.org/index.php/Language_Learning_and_Age Baker, F n.d., Children learning a second language, Kidspot, viewed 05 March 2015, http://www.kidspot.com.au/schoolzone/General-Children-learning-a-second-language+4095+396+article.htm Clarke, P 2009, Supporting children learning English as a second language in the early years (birth to sic years), VCAA, viewed 05 March 2015, http://www.vcaa.vic.edu.au/documents/earlyyears/supporting_children_learning_esl.pdf Costandi, M 2014, ‘Am I too old to learn a new language?’, The Guardian, 13 September, viewed 26 February 2015, http://www.theguardian.com/education/2014/sep/13/am-i-too-old-to-learn-a-language Ford, C 2014, †Children should start learning languages at age three†, The Telegraph, 10 October, viewed 05 March 2015, http://www.telegraph.co.uk/education/educationopinion/11151726/Children-should-start-learning-languages-at-age-three.html Henry, J 2012, ‘Foreign language to be compulsory from age seven’, The Telegragh, 09 June, viewed 26 February 2015, http://www.telegraph.co.uk/education/primaryeducation/9321651/Foreign-languages-to-be-compulsory-from-age-seven.html Mclaughlin, B 1992, Myths and misconceptions about second language learning: what every teacher needs to unlearn, University of California, viewed on 26 February 2015, https://www.usc.edu/dept/education/CMMR/FullText/McLaughlinMyths.pdf Merritt, A 2013, ‘Are children really better at foreign language learning?’, The Telegraph, 18 September, viewed 05 March 2015, http://www.telegraph.co.uk/education/educationopinion/10315238/Are-children-really-better-at-foreign-language-learning.html Schartwen, E 2008, ‘Aussie kids ‘need a second language’’, Sydney Morning Herald, 23 May, viewed 26 February 2015, http://www.smh.com.au/national/aussie-kids-need-a-second-language-20080522-2hey.html Schouten, A 2009, The critical period hypothesis: support, challenge, and reconceptualization, Kanda University of International Studies, viewed 04

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