In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope though these entities were quantitatively small; the word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which referred to the "infinity-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object, smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.
The concept of infinitesimals was introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion; the 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors; the method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus.
He exploited an infinitesimal denoted 1/∞ in area calculations. The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds for the infinite numbers and vice versa; the 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, in defining an early form of a Dirac delta function; as Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed nonstandard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.
The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it is not popular to talk about infinitesimal quantities. Present-day students are not in command of this language, it is still necessary to have command of it. The notion of infinitely small quantities was discussed by the Eleatic School; the Greek mathematician Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1... and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains infinitesimal members; the English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections.
The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his Treatise on the Conic Sections, Wallis discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞; the concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus; the conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval. Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
Prior to the invention of calculus mathematicians were able to calculate tangent lines using Pierre de Fermat's metho
The Ming-Ai Institute is the executive arm of the Ming-Ai Association, established in 1993 to promote Chinese culture locally and deliver cultural exchanges between the UK and Greater China. Operating from Denver House near Bounds Green tube station, the Ming-Ai Institute offers a number of short courses and delivers a range of undergraduate and postgraduate courses in Memorandum with Middlesex University; the Ming-Ai Institute hosts and exhibits information about British Chinese cultural Heritage through the British Chinese Heritage Centre, a cyber centre dedicated to on-going and past heritage projects conducted by the Ming-Ai Institute. The institute has delivered a variety of professional and vocational courses which include the following: languages, including Japanese, Cantonese and, Mandarin. Theresa Wai Han Shak is the founder of the Ming-Ai Institute. Theresa's origins in Mainland China, inherited wealth, connections to the Catholic Church and passion for contemporary education allowed Shak to become influential in educational reform and UK-China relations.
Understanding her journey and mission provides insights to the institutes origins and purpose. To fulfill the mission of the Ming-Ai Institute numerous projects have been delivered towards cultural exchange between the UK and China and promote Chinese culture locally. Projects include a number of oral histories which form the basis for a series of research publications, community events, training events and workshops; the training of volunteers and staff is an important measure of success for each project. The institute is a placement partner of Goldsmiths, University of London. Funded by the Heritage Lottery Fund, British Chinese Armed Forces is an ongoing project launched in June 2015; the project is a four-year undertaking in collaboration with Regent's University London for the creation of a cultural-historical archive documenting the contributions made by people of Chinese descent to the British Armed Forces. In partnership with the National Army Museum, the institute will collect the stories about historical items.
The project has been mentioned in The Huffington Post. Elizabeth Ride provided the Ming-Ai Institute with her account of her father, Sir Lindsay Ride's career in British Hong Kong with the British Army; the recording, archived at the British Chinese Heritage Centre speaks of how Ride came to form the British Army Aid Group. A Brigadier stationed in Hong Kong. Brigadier Christopher Hammerbeck a former Deputy Commander and Chief of Staff of British Forces Overseas Hong Kong. Commodore Peter Melson The final commodore of HMS Tamar in Hong Kong. Funded by the Heritage Lottery Fund, British Chinese Work Force Heritage was a three-year project launched in 2012 with provision to explore the contributions made by British Chinese to the London workforce over the past 150 years during which Ming-Ai Institute trained 12 interns, published 89 oral histories and five articles in partnership with the City of London, Haringey Council, London Metropolitan Archives, Islington Heritage, National Army Museum, Regent's University London, Horniman Museum & Gardens, St Micheal's Catholic College, University College London, Middlesex University London, K&L Gates and City of Westminster Libraries.
The British Chinese Workforce Heritage project was written about in the South China Morning Post. Funded by the Heritage Lottery Fund, the British Chinese Food Culture project was launched in 2011 in order to identify the changes in British Chinese Cuisine from the original recipes derived from Greater China. A key focus is how the availability of ingredients caused Chinese restaurants to adapt their dishes and explores how the reintroduction of original ingredients allows restaurants to deliver greater authenticity. From 2016 the new GCSE in Food Preparation & Nutrition will be taught in British schools. In preparation the Ming-Ai Institute in partnership with Chinese manufacturer Lee Kum Kee founded a project to promote Chinese cuisine in British schools; the project will last for five years with the objective of reaching 280 schools and conduct 40 or more teacher training workshops. Funded by the Heritage Lottery Fund, East West Festive Culture was a two-year project which started in October 2008.
The project aimed to explore analogous festivals in Chinese cultures. Tracing 150 years' of British Chinese festive celebration in London the project cast light on a three demographics. Firstly Chinese people who emigrated to the UK in their early life, secondly couples with a non-Chinese partner and British born Chinese; each of these groups was selected to shed light to both the contrast and similarities between Eastern and Western culture. In October 2009 the Ming-Ai Institution used funding from UK Government Transformation Fund to undertake The Evolution and History of British Chinese Workforce project; this was the second oral histories project undertaken by the Ming-Ai Institute. Funded by the Electoral Commission the Making Chinese Votes Count project was managed by a consortium consisting of both the Ming-Ai Institute and
Capital controls were introduced in Greece in June 2015, when Greece's government came to the end of its bailout extension period without having come to an agreement on a further extension with its creditors and the European Central Bank decided not to further increase the level of its Emergency Liquidity Assistance for Greek banks. As a result, the Greek government was forced to close Greek banks for 20 days and to implement controls on bank transfers from Greek banks to foreign banks, limits on cash withdrawals, to avoid an uncontrolled bank run and a complete collapse of the Greek banking system; the capital controls were minimized until their complete removal on the 1st of September 2019. In September 2015, certain aspects of the imposed capital controls were relaxed. Four months after capital controls were imposed on 28 June 2015, two important modifications were published by the government: while still limiting withdrawals to €420 per week, account holders could withdraw the whole sum in one transaction instead of up to only €60 per day, thus reducing the amount of time spent queueing at the banks and ATMs, furthermore, up to 10% could be withdrawn from funds deposited in Greece from abroad.
To minimize the impact on tourism, people with foreign credit cards such as tourists could withdraw through ATMs higher amounts of cash. Other changes allowed time deposits to be terminated prematurely to cover real estate purchases and living expenses up to €1800 per month. Furthermore, resident Greeks making payments or remittances to foreign banks in which they had accounts in their name had the €1800 limit removed and businesses making payments abroad could send up to €5000 per day per client without seeking special permission. Still, the impact on business was dramatic as many exporters could not import the raw materials they needed for production as payments above the limits had to be approved by special committees who could approve imports of only €20 million per day in total with priority on medicines and food. Six months into capital controls saw a further easing of the rules. While full or partial premature repayment of loans was still not allowed, the premature repayment of loans could be made by loan holders either through funds from abroad or by taking out a new loan to cover the amounts due on previous outstanding loans.
Moreover, although still limited to €420 of withdrawals per week, special exemptions could be made to pay administration fees and debts to the state. In December 2015, the finance ministry signed off on measures that were designed to bolster the ailing Greek stock markets. Investors could now use "old money", to make transactions on the stock market in addition to the allowed "new money" from abroad or from the sale of stocks or other financial assets such as dividends from stocks. Following the first successful review of Greece's 3rd bailout memorandum of August 2015 by the state's creditors in June 2016 and in its "11th ministerial decision on capital controls" since imposition, on 22 July 2016 the Greek government's Finance Ministry announced that "new cash" deposited in Greek banks from abroad would be free of any limit on withdrawals to encourage more of the money withdrawn in the run-up to capital controls to find its way back into the Greek banking system which had witnessed a €4.5 billion increase in deposits to €127 billion in the two months following the successful review as market jitters calmed.
The decision raised the limit on "old cash" withdrawals to €840 per two weeks instead of €420 per week. And in addition to first-year university students moving away from home to study and, given permission to open bank accounts, Erasmus students and pensioners living abroad were given permission to open new bank accounts in Greek banks. Still, the limit on cash for individuals travelling abroad remained at €2000; the effects of capital controls changed customer payment habits. Since the controls on withdrawals did not apply to the use of credit/debit cards to make purchases in Greek retail outlets, the average use of credit card transactions jumped from 4.5% to 19.5% in a short time and up to 35% in supermarket transactions with more than 50% of people saying according to the Bank of Greece that they used their cards more than in the past, all of which left a paper trail that the Greek government was keen to encourage in its combat against tax evasion, despite the fact that the general use of credit transactions is believed to drive up the overhead for businesses and increase the price of goods and services for consumers.
Before capital controls, in 2014, on average every Greek citizen used cards for transactions, not including ATMs, on average 8 times per year against a 93-time European Union average, while in 2015 the numbers rose to 20 times per year per citizen and representing €9 billion, double that of the previous year. Despite positive pronouncements by bank insiders and the government that capital controls could be soon lifted or at least by the end of 2016, more than a year after their imposition, capital controls were still in place, unlike the example of Cyprus after the Cypriote crisis precipitated in March 2013 and who had eliminated their capital controls in April 2014 just over one year after the crisis had erupted, it must, however, be emphasized that the Cypriote crisis was more of a banking crisis than a state debt crisis and therefore not analogous. Fears of a complete