Saturday, September 14, 2019

Lacsap’s Triangle

1 Introduction. Let us consider a triangle of fractions: Obviously, the numbers are following some pattern. In this investigation we will try to explain the theory behind this arrangement and to find a general relation between the element’s number and its value. The pattern above is called a Lacsap’s Triangle, which inevitably hints at its relation to another arrangement – Pascal’s Triangle (as Lacsap appears to be an anagram of Pascal). The algorithm behind it is very simple: each element is the sum of the two elements above it.However, if we represent a triangle as a table (below), we will be able to notice a pattern between an index number of an element and its value: column column column column column column column 2 0 1 2 3 4 5 row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1 row 6 1 6 15 20 15 6 6 1 It seems important to us to stress several points that this table makes obvious: ? the number of elements in a row is n + 1 (where n is an index number of a row) ? the element in column 1 is always equal to the element in column n – 1 ? herefore, the element in column 1 in every row is equal to the number of a given row. Now when we have established the main sequences of a Pascal’s triangle let us see how they are going to be expressed in a Lacsap’s arrangement. We also suggest looking at numerators and denominators separately, because it seems obvious that the fractions themselves can’t be derived from earlier values using the progressions of the sort that Pascal uses. Finding Numerators. Let’s begin with presenting given numerators in a similar table, where n is a number of a row. n=1 1 1 n=2 1 3 1 n=3 1 6 6 1 n= 4 1 0 10 10 1 n=5 1 15 15 15 15 1 3 Although the triangles appeared similar, the table demonstrates a significant difference between them. We can see, that all numerators in a row (except 1’s) have the same value. Therefore, they do not depend on oth er elements, and can be obtained from a number of row itself. Now a relationship we have to explore is between these numbers: 1 1 2 3 3 6 4 10 5 15 If we consider a number of row to be n, then n=1 1=n 0. 5 2 n 0. 5 (n +1) n n=2 3 = 1. 5n 0. 5 3 n 0. 5 (n +1) n n=3 6 = 2n 0. 5 4 n 0. 5 (n +1) n n=4 10 = 2. 5 n 0. 5 5 n 0. 5 (n +1) n n=5 15 = 3n 0. 6 n 0. 5 (n +1) n Moving from left to right in each row of the table above, we can clearly see the pattern. Dividing an element by a row number we get a series of numbers each one of them is 0. 5 greater than the previous one. If 0. 5 is factored out, the next sequence is {2; 3; 4; 5; 6}, where each element corresponds to a row number. Using a cyclic method, we have found a general expression for the numerator in the original triangle: If Nn is a numerator in a row n, then Nn = 0. 5(n + 1)n = 0. 5n2 + 0. 5n Now we can plot the relation between the row number and the numerator in each row.The graph of a parabolic form begins at (0; 0) and co ntinues to rise to infinity. It represents a continuous function for which D(f) = E(f) = (0; ); 4 Using a formula for the numerator we can now find the numerators of further rows. For example, if n = 6, then Nn = 0. 5 62 + 0. 5 6 = 18 + 3 = 21; if n = 7, then Nn = 0. 5 72 + 0. 5 7 = 24. 5 + 3. 5 = 28; and so forth. Another way of representing numerators would be through using factorial notation, for obviously Numeratorn = n! Now let’s concentrate of finding another part of the fraction in the triangle. Finding Denominators.There are two main variables, that a denominator is likely to depend on: ? number of row ? numerator To find out which of those is connected with the denominator, let us consider a following table: column 1 column 2 column 3 column 4 column 5 column 6 5 row 1 1 1 row 2 1 2 1 row 3 1 4 4 1 row 4 1 7 6 7 1 row 5 1 11 9 9 11 1 It is now evident, that a difference between the successive denominators in a second column increases by one with each iteration: {1; 2 ; 4; 7; 11}, the difference between elements being: {1; 2; 3; 4}. So if the number of row is n, and the denominator of the second column is D, then D1 = 1D2 = 2 D3 = 4 etc; then Dn = Dn-1 + (n – 1) = (n-1)! + 1; If we now look at the third column with a regard to a factorial sequence, a pattern emerges: In the series {1; 1; 2; 3; 4; 5; 6; 7;†¦ ; }, if d is the denominator of the third column, then: d3 = 1 + 1 + 2 = 4 d4 = 1 + 2 + 3 = 6 d5 = 2 + 3 + 4 = 9 dn = (n – 2)! + 3; To check the consistency of this succession, we will continue with the study of the fourth column. By analogy, the result is as follows: Denominatorn = (n – 3)! + 6 (where n is a number of row) Therefore, it can be represented as follows:Column 2 (n-1)! +1 Column 3 (n-2)! +3 Column 4 (n-3)! +6 It is now clear, that numbers inside the brackets follow the (c – 1) (where c is the number of column), and the numbers outside are in fact the numerators of the row of the previous index num ber (comparing to the column). Therefore, a general expression for the denominator would be Dn = (n – (c – 1))! + (c – 1)! 6 where Dn is a general denominator of the triangle n is a number of row c is the number of column Now we can use a formula above to calculate the denominators of the rows 6 and 7. column 2 column 3 olumn 4 column 5 column 6 row 6 (6 – 1)! + 1 = 16 (6 – 2)! + 3 = 13 (6 – 3)! + 6 = 12 (6 – 4)! + 10 = 13 (6 – 5)! +15 =16 row 7 (7 – 1)! + 1 = 22 (7 – 2)! + 3 = 18 (7 – 3)! + 6 = 16 (7 – 4)! + 10 = 16 (7 – 5)! +15 =18 column 7 (7 – 6)! + 21 = 22 Fusing these value with the numerators from the calculations above, we get the 6th and the 7th rows of the Lacsap’s triangle: Row 6: 1; ; ; ; ; ;1 Row 7: 1; ; ; ; ; ; ;1 If we now let En(r) be the (r + 1)th element in the nth row, starting with r = 0; then the general statement for this element would be: En(r) =Conclusion. To check the validity and limitations of this general statement let us consider the unusual circumstances: first of all, will it work for the columns of ones (1st and last column of each row)? if n = 4 r = 0, then En(r) = =1 if n = 5 r = 5, then En(r) = =1 7 therefore, the statement is valid for any element of any row, including the first one: En(r) = =1 However, obviously, the denominator of this formula can not equal zero. But as long as r and n are both always positive integers (being index numbers), this limitation appears to be irrelevant.If the numeration of columns was to start from 1 (the 1st column of ones), then the general statement would take the form of: En(r) = 8 Bibliography: 1) Weisstein, Eric W. â€Å"Pascal's Triangle. † From MathWorld–A Wolfram Web Resource. http:// mathworld. wolfram. com/PascalsTriangle. html 2) â€Å"Pascal’s Triangle and Its Patterns†; an article from All you ever wanted to know http:// ptri1. tripod. com/ 3) Lando , Sergei K.. â€Å"7. 4 Multiplicative sequences†. Lectures on generating functions. AMS. ISBN 0-8218-3481-9

Friday, September 13, 2019

Assessment of the Idea of Liberty As Illustrated By Hannah Arendt In, What Is Freedom

Assessment of the Idea of Liberty As Illustrated By Hannah Arendt In, What Is Freedom The concept of freedom is an abstract one that is only realized when a person sets it into motion. In What is Freedom, Hannah Arendt challenges the widespread belief that liberty extends from the freedom of thought and will. Arendt emphasizes that actions performed unhinged from consequences are true bouts of freedom. What is Freedom dispels the accustomed definition found in government, textbooks, and marketplace and replaces it with freedom as a gateway of creativity and boundlessness. The common sense notion of freedom has been etched into the framework of our society. The â€Å"Land of the Free† otherwise known as the United States of America, etches the entitlement to our life, liberty, and the pursuit of happiness as the focal point of its’ Constitution. Similarly, the Merriam-Webster dictionary defines freedom as â€Å"the state of being free† and â€Å"a political right† ( In economics, the term laissez-faire, the abstention by governments from interfering in the workings of the free market† is the quintessential cornerstone of one of our most defining national traits, capitalism ( All three of these definitions stress, in one form or another, that freedom is possessive. We possess rights that protect us from an oppressive government, slavery, or even the business-incentive crushing Communism. The most general of these meanings â€Å"the state of being free† only serves to a dd intangibility and uncertainty to an already scantly defined concept. This common sense notion of freedom umbrellas the different ways a person can make decisions. These can be the rationality of a person to select a choice, to want/need to have ownership of property, or incentive work towards a cause of interest. Despite this general understanding, Arendt argues against these concepts. While â€Å"we hold human freedom to be a self-evident truth† Arendt defines the common sense notion just as the inward space into which men may escape from external coercion and feel free† (Arendt 647, 649). Freedom in its pure form is an abstract concept. Since we cannot empirically measure freedom by the number of people emancipated from slavery, nor can we quantify its’ workings through calculating marginal utility, the crux comes from establishing an encompassing operational definition. Arendt developed a solution stating that for an â€Å"Action to be free, it must be free from motive on one side, from its intended goal as a predictable effect on the other† (Arendt 653). Thus for the infinite number of possibilities that can result from making a choice or an action, as long as the person makes a choice unbeknownst of any external influence, they are free. For example choosing between multiple choice options on an AP exam does not entail freedom if the test taker reads the question and selects the answer that best fits, even if there exist a potential margin of error. Contrariwise, if the student spontaneously selects a choice based on intrinsic preferences for an answe r letter, ignoring the terms of the question and the answers, he has attained freedom. In this way the answer â€Å"as it is free is neither under the guidance of the intellect nor under the dictate of the will† (Arendt 654). If will and intellect are engaged, however, the person is guided towards a solution logically and thus freedom is not present. The student is using goal setting, or aim. Since â€Å"Aim is not a matter of freedom, but of right or wrong judgment† if the student answers option C because the question leads them to the logical conclusion that the text in C is correct they practiced aim not freedom (Arendt 653). Arendt’s conception of freedom is based on the concept of principles. These principles â€Å"such as honor or glory, or love of equality . . . inspire. . . [and become] fully manifest only in the performing act itself† (Arendt 654). This inspiration is the reason for freedom’s volatility. When a person exerts an action in a free state the person is guided by this burst of inspiration from the principle to select one out of an unlimited number of outcomes. Inspiration differs from that of an aim because is not fully realized until the action is completed. Additionally principles lose â€Å"nothing in strength or validity through execution† unlike goals which vanish once they are completed (Arendt 654). Since principles are everlasting sources of inspiration, they outline the creative freedom that exists within man. Arendt arrived at the conclusion that the ability to act freely, guided only by principles, is the reason that art exists. Humans exercise unrestricted talents with unlimited outcomes guided by undefined inspiration when creating art. This is why â€Å"the element of freedom, certainly present in the creative arts, remains hidden† (Arendt 655). Although the end result of art is often guided by a motive such as recognition or money, the artist that created it was guided freely without any bounds. The effect of inspiration on a person becomes the axiom that â€Å"freedom is a worldly reality, tangible in words that can be heard, in deeds which can be seen, and in events which are . . . incorporated into the great storybook of human history† (Arendt 656). Arendt imposes a new meaning on the liberty of humans. Unlike the common sense notion where freedom exists in the forms of; ownership of property, the right to unalienable rights, and options to choose, a person practicing Arendt’s notion of freedom is not restricted by the choices in front of them. By letting go of your ephemeral motives, by replacing it with inexhaustible principles you unlock a freedom that has no bounds. Instead of having a choice from options, you can pick an undefined number of other options that are not listed. Conclusively Arendt believes that, freedom is not the right to choose your own adventure. Freedom is the ability to create it.

Thursday, September 12, 2019

Relationship between two currenies Essay Example | Topics and Well Written Essays - 1000 words

Relationship between two currenies - Essay Example A country can determine the price of a currency against another currency in two ways, which include fixed and floating exchange rate. Fixed or pegged exchange rate is rate of currency determined and maintained by Central Bank. â€Å"In order to maintain local exchange rate, the central bank buys and sells it own currency in the foreign exchange market in return for the currency against which it is pegged† (Mano, 2010). On the other hand, floating exchange rate is determined by market a force, which means levels of supply and demand of the currency, decides at which price the currency will be sold in foreign exchange market. There are multifarious factors, which determine the exchange rate. This includes higher interest rate that would attract the foreign financial capital inflow in the local country and foreigners demand for local currency escalates resulting in appreciating exchange rate. Another factor is economic health, which means foreign investors are likely to invest in countries with high positive indicators like inflation growth and debt burden rather than in economies, which are weak. Exchange rate is also quite susceptible to shocks and speculation. In addition, money markets are liquid so exchange rates are responsive to sudden shocks (Madura, 2008). Currency rates even move because of speculative investments or if brokers trade them as per their expectations of exchange rate. One other important factor is government or central bank intervention, which we already discussed under fixed exchange rate. The two major strong currencies of the world are dollar and Euros. It is not possible to conclude the exchange rate discussion without mentioning about them (Madura, 2008). There are different theories by which economists explain the general movements in exchange rate. However, none of them is strong enough to describe the exact movements so for; therefore, they explain these movements with the term â€Å"random walk pattern†. It is impossi ble to overlook the exchange rate between dollar and yen when discussing reasons behind the general movements in exchange rates. This is because of the appreciating yen against dollar that has been under maintenance since last 15 years, thus, it is important to explore the root causes behind this appreciation (Madura, 2008). Observing the exchange rate for last seven months of yen against one dollar was 85, 84, 81,82,83,82, and 83 in August to February period. Hence, on average, yen has appreciated from August 2010 to January 2011 but it bounced back against dollar on February 15, 2011 at rate of 83.7972. The foremost thing to consider when deciding on the reasons of general movements in exchange rate is that exchange rate is just the price of one currency in terms of another. If yen is strong against dollar it means it is stronger relative to dollar that is dollar is weak and yen is not strong in itself (Madura, 2008). The major reason for weak dollar is due to its global financial crisis and most probably risk of â€Å"second dip’ which means second recession. Apart from these crises, President Obama is in a flabby situation because his mid- term congress elections are round the corner. Hence, these shaky situations are resulting in a search for â€Å"safe haven† as investors are finding yen as a stable currency as compared to dollar or euro. This results in yen appreciation (Madura, 2008). There are not only direct reasons, which result in appreciation of yen

Global Warming speech by Marcus Gibson Essay Example | Topics and Well Written Essays - 500 words

Global Warming speech by Marcus Gibson - Essay Example The speech has been delivered where it tends to be more convenient to pass the message and the audience is encouraging.Furthermore, it encourages everyone to be walking shorter distances rather than using vehicles which pollute the environment. The speech has not ignored the negative challenges but has acknowledged all the aspects. It has tried to lead the audience in understanding the speech being delivered. Compelling the audience in practicing the safety measures against preventing global warming has been emphasized. This is by encouraging them to do the very simple thing like walking for short distances rather than using their vehicles. Furthermore, the speech is more obliged and focuses on achieving the best from the audience. On delivering the speech, the person engaged the audience by using eye contact and movements. This is with the fact that they the audience may provide visual support and encouragement during the delivery of the speech as they are considered to be the focal point. In addition, the speech on Global warming has been delivered in the manner that at that time every person has been willing to fight and prevent it. The writer focused on using rhetoric questions in delivering the speech. For instance, â€Å"If we don’t prevent global warming by ourselves, who will?†This made the audience to be livelier and contribute by providing good listening skills. Moreover, the speech being delivered focused on the key issues making it be precise thus delivering the intended message to the audience. The persuasive attempt in delivering the speech was more effective.

Wednesday, September 11, 2019

Marketing led management(Apple's iphone) Essay Example | Topics and Well Written Essays - 2750 words

Marketing led management(Apple's iphone) - Essay Example The study will be focused upon the UK based company Telefonica,popularly known as O2.It has been in venture with Apple to promote iPhone in the UK market.The brand O2 caters the services in the telecommunication sector offering integrated mobile,fixed and broadband services in the UK,Germany,Ireland,Slovakia and Czech Republic In the year 2007, Apple launched its product iPhone with O2 in the UK market. Apple iPhone was offered in different tariffs with variety of handsets in the UK market (Guardian News and Media Limited, 2011). In order to launch iPhone in the European market, Apple needs to have a contract with the service provider. They had options in the European market. There were UK’s O2, France’s Orange and Germany’s T-Mobile. There were rumours regarding the revenue sharing percentage deal between Apple and others. After the plethora of negotiations, the UK’s O2 was finalised for the deal to market Apple’s product . Strategic Analysis The d eal was finalised between the UK’s O2 and Apple for the distribution of the Apple’s iPhone in the UK market. The deal concentrated upon the revenue sharing percentage that O2 would provide to Apple from the iPhone sales. The significance of the deal to O2 can be justified by the strategic analysis through the SWOT evaluation. SWOT Analysis Strengths O2 is known as the successful network operator in the UK market. The awareness of the brand is huge among the target customer group. There are more than 300 stores of O2 in the UK market (Franchise Development Services Ltd, 2011). Telefonica has been operating for more than 85 years in the UK telecommunication industry and has a consistency growth rate and is acquainted with their target consumers’ need and wants. By the year 2009 it had about 265 million customers operating in 25 countries. Telefonica provides integrated ICT solutions to all segments of customers that can be demonstrated in a graphical format below. Telefonica has diverse operations and generates revenue from different market operations, which build up a strong financial position of the company (Telefonica, 2010). (Telefonica, 2010). Telefonica has been able to increase its consumer base by 2009 and it is still on the rise. The company achieved a 9 % growth in the number of consumers by the end of 2009. The expansion in the consumer base will generate more revenues for the company. With the increasing number of consumers there is more possibility of increase in the sales of iPhone. This will generate more revenues for Apple (Telefonica Europe plc, 2009). Apple is one of the leading brands in the global market and is known for its iPhone. The newly launched product of Apple is the ‘iPhone 4’ in the UK market. O2 provides maximum advantage to every class of users at an affordable rate (PR Wall Street, 2010). The mobile network of Telefonica covers 99% of the UK populace. O2’s 3G network covers over 80% of the UK’s populace. The best selling device of Telefonica O2 UK is the iPhone that has more than 2 million sales (Telefonica, 2010). Weaknesses The UK consumers had the facility of 3G phones before the launch of the iPhone in the UK market. They were already using the 3G technology. There were problems related to the offering of the call rates. The call rates of the iPhone deal allowed 200 inclusive minutes for ?35 whereas the competitors provided 750 inclusive minutes with a free phone. The consumers were apprehensive about the contract related to iPhone. The main concern was regarding the up-gradation of the phone, the device itself as it has non-removable battery and limited battery life. The consumers wanted to upgrade their iPhone 3G as it faced the problem of battery life. This problem was a concern for Telefonica O2 UK against their endeavour to increase the sales and get

Tuesday, September 10, 2019

Human Factors in aircraft operations in unfamiliar airfields Research Paper

Human Factors in aircraft operations in unfamiliar airfields - Research Paper Example As per a study, about 80% of the aircraft accidents happen during either landing or take-off of jets, and, thus, such accidents occur at close proximaty to an airport. Pilots may resort to unsafe acts, which may culminate into aircraft accidents due to human errors and violations of standard norms. This research essay will focus on the human factors witnessed by the aircrew in unfamiliar airfields and suggest ways and means to avoid and overcome fallacies that may lead to aircraft accidents. Introduction Some of the deadliest aircraft accidents happened not in the air but on the runways of airports. About 583 air passengers were killed in 1977 in the Canary Islands, when two jumbo jets crashed on a runway. A runway incursion accident in Chicago’s Midway airport in 1991 killed 34 air passengers. Unfamiliar braking system resulted in an accident in Chicago’s midway airport in December 2005. When a pilot took a wrong runway for takeoff in Lexington airstrip, about 49 air p assengers were killed in August 2006. When the captain was not able to manage for a gusty and strong cross wind during takeoff, the aircraft departed to the side of the runway in December 2008 at Denver International Airport. (NTSB 2012). Airport plays a pivotal role as it is the place where the aircraft is landing, taking off, parking, and taxing. According to ICAO (International Civil Aeronautics Organization), an aviation accident is one, which is happening of a fatal event during the operation of an aircraft. An accident in an airport may involve the following impacts namely as the accident in an airport during landing may involve severe damage to the atmosphere which includes the loss of property or life, and it may also create an acute social affliction. Further, airport accident while landing in a new location may emanate from various environmental factors like failure of equipment, human error and other associated risk elements. (Czerny, 2010, p.4-3). This research essay wil l conduct analysis about the issues faced by the pilots while landing at unfamiliar airfields and will make suggestion how to overcome the same. Issues Faced by the Pilots in Unfamiliar Airfields while Landing or Takeoff The ultimate safety of the aircraft and passengers are entrusted to the captain of the flight. There is a need on the part of the captain to understand strengths and Achilles' heel of each of the crew members and should be able to assign actions whenever it is needed. Under the Crew Resource Management (CRM), flight crew security is covered. In resolution of any probable security incident, captain is needed to treat and respect all crew members equally and properly. Proper training should be given to crew members to develop a strong state of mental ability for handling any critical situation. Each and every crew member should build up the buoyancy to be aware what is to be done in an emergency scenario and to react suitably when an incident occurs. (Williams & Wallt rip, 2004, p354). As per a study, about 80% of the jet aircraft accidents happen during either landing or take-off of flights, and thus such accidents are occurring at the proximate of an airport. Hence, there is a heightened peril of serious injury or death both to passengers and to the nearby communities residing close to the airport. If we analyze the happening of an accident during landing or take –off, one can find many reasons like the type of aircraft involved, the airport infrastructures, human factors and metrological conditions. Wong et

Monday, September 9, 2019

Excavation of an early Christian catacomb- Catacomb of Priscilla Research Paper

Excavation of an early Christian catacomb- Catacomb of Priscilla - Research Paper Example One of the most phenomenon paintings presents the oldest-known image of Mary the mother of Jesus. The image depicts her nursing the Infant Jesus. This painting that emphasizes on the Key figure in Christianity-Jesus-clearly explains the significant role of Jesus in the Christian history. The significant of the image is that it gives an idea that Christianity has been in existence as early as the 2nd century. This gives confidence to Christians of the existence of their faith based on the painting that were created very many years ago. Further, the painting of Jesus and her mother serves as an evidence of some biblical perspectives. In this case, the painting can be taken to represent some fact as the artist who put together the piece of art must have been existing during the time of Jesus. The catacomb of Priscilla also has a depiction of the Annunciation. The piece of art in the catacomb explains the incident of the announcement of the manifestation by the angel Gabriel to Mary the mother of Jesus. The piece of art is represented as a ceremony festival commemorating this incident. According to the painting, an angle is seen appearing to Mary to make announcement of the birth of Jesus. Similarly, this piece of art found in the Catacomb of Priscilla is significant as it gives confirmation that the incident took place according to the records in the Bible. Art, being a reflection tool in the society, clearly shows that the story of Jesus was very much told during the time. The societies in the 2nd Century totally embraced the Kingship of Jesus and hence the many art work on the subject, (Neiman 15) In our societies today, every person is looking for proof and evidence on various incidents that are explained in the Bible. As such, various sections of the Bible have been challenged by scholars and philosophers concerning their