Probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
Frisch family
Frisch is a Norwegian family of German origin, of which many members were associated with the Kongsberg Silver Mines for several centuries since the 17th century. In the 19th and early 20th centuries, family members were noted as goldsmiths in Oslo, its most famous member is the first Nobel laureate in economics. It is descended from Christopher Frisch, who came from Germany to work for the Kongsberg Silver Mines in Norway, he was the father of Petter Christophersen Frisch. His sons were Christopher Pettersen Frisch. Paul Pettersen Frisch was married to Anne Antoniusdatter Nolt, daughter of Antonius Nolt and Barbro Hansdatter Schröder, a daughter of Hans Andersen Schröder. Paul Frisch and Anne Nolt were the parents of Ole Povelsen Frisch. Ole Povelsen Frisch was the great-grandfather of goldsmith in Christiania Antonius Povelsen Frisch, the father of goldsmith Anton Frisch and grandfather of the economist and Nobel laureate Ragnar Frisch; the Frisch Centre in Oslo and the Frisch Medal in economics are both named for Ragnar Frisch.
Ragnar Frisch was the maternal grandfather of the television presenter Nadia Hasnaoui. Huhnhäuser, Die deutsche Einwanderung in Kongsberg, Oslo, 1944 Såtvedt, Sandsværs historie, 1991
Italy
Italy the Italian Republic, is a country in Southern Europe. Located in the middle of the Mediterranean Sea, Italy shares open land borders with France, Austria and the enclaved microstates San Marino and Vatican City. Italy covers an area of 301,340 km2 and has a temperate seasonal and Mediterranean climate. With around 61 million inhabitants, it is the fourth-most populous EU member state and the most populous country in Southern Europe. Due to its central geographic location in Southern Europe and the Mediterranean, Italy has been home to a myriad of peoples and cultures. In addition to the various ancient peoples dispersed throughout modern-day Italy, the most famous of which being the Indo-European Italics who gave the peninsula its name, beginning from the classical era and Carthaginians founded colonies in insular Italy and Genoa, Greeks established settlements in the so-called Magna Graecia, while Etruscans and Celts inhabited central and northern Italy respectively; the Italic tribe known as the Latins formed the Roman Kingdom in the 8th century BC, which became a republic with a government of the Senate and the People.
The Roman Republic conquered and assimilated its neighbours on the peninsula, in some cases through the establishment of federations, the Republic expanded and conquered parts of Europe, North Africa and the Middle East. By the first century BC, the Roman Empire emerged as the dominant power in the Mediterranean Basin and became the leading cultural and religious centre of Western civilisation, inaugurating the Pax Romana, a period of more than 200 years during which Italy's technology, economy and literature flourished. Italy remained the metropole of the Roman Empire; the legacy of the Roman Empire endured its fall and can be observed in the global distribution of culture, governments and the Latin script. During the Early Middle Ages, Italy endured sociopolitical collapse and barbarian invasions, but by the 11th century, numerous rival city-states and maritime republics in the northern and central regions of Italy, rose to great prosperity through shipping and banking, laying the groundwork for modern capitalism.
These independent statelets served as Europe's main trading hubs with Asia and the Near East enjoying a greater degree of democracy than the larger feudal monarchies that were consolidating throughout Europe. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science and art. Italian culture flourished, producing famous scholars and polymaths such as Michelangelo, Leonardo da Vinci, Raphael and Machiavelli. During the Middle Ages, Italian explorers such as Marco Polo, Christopher Columbus, Amerigo Vespucci, John Cabot and Giovanni da Verrazzano discovered new routes to the Far East and the New World, helping to usher in the European Age of Discovery. Italy's commercial and political power waned with the opening of trade routes that bypassed the Mediterranean. Centuries of infighting between the Italian city-states, such as the Italian Wars of the 15th and 16th centuries, left the region fragmented, it was subsequently conquered and further divided by European powers such as France and Austria.
By the mid-19th century, rising Italian nationalism and calls for independence from foreign control led to a period of revolutionary political upheaval. After centuries of foreign domination and political division, Italy was entirely unified in 1871, establishing the Kingdom of Italy as a great power. From the late 19th century to the early 20th century, Italy industrialised, namely in the north, acquired a colonial empire, while the south remained impoverished and excluded from industrialisation, fuelling a large and influential diaspora. Despite being one of the main victors in World War I, Italy entered a period of economic crisis and social turmoil, leading to the rise of a fascist dictatorship in 1922. Participation in World War II on the Axis side ended in military defeat, economic destruction and the Italian Civil War. Following the liberation of Italy and the rise of the resistance, the country abolished the monarchy, reinstated democracy, enjoyed a prolonged economic boom and, despite periods of sociopolitical turmoil became a developed country.
Today, Italy is considered to be one of the world's most culturally and economically advanced countries, with the sixth-largest worldwide national wealth. Its advanced economy ranks eighth-largest in the world and third in the Eurozone by nominal GDP. Italy owns the third-largest central bank gold reserve, it has a high level of human development, it stands among the top countries for life expectancy. The country plays a prominent role in regional and global economic, military and diplomatic affairs. Italy is a founding and leading member of the European Union and a member of numerous international institutions, including the UN, NATO, the OECD, the OSCE, the WTO, the G7, the G20, the Union for the Mediterranean, the Council of Europe, Uniting for Consensus, the Schengen Area and many more; as a reflection
Rockefeller Foundation
The Rockefeller Foundation is a private foundation based at 420 Fifth Avenue, New York City. It was established by the six-generation Rockefeller family; the Foundation was started by Standard Oil owner John D. Rockefeller, along with his son John D. Rockefeller Jr. and Senior's principal oil and gas business and philanthropic advisor, Frederick Taylor Gates, in New York State on May 14, 1913, when its charter was formally accepted by the New York State Legislature. Its stated mission is "promoting the well-being of humanity throughout the world." Rockefeller Foundation's activities have included: Financially supported education in the United States "without distinction of race, sex or creed" Helped establish the London School of Hygiene and Tropical Medicine in the United Kingdom. Construction of the Kaiser Wilhelm Institute's Institute for Brain Research with a $317,000 grant in 1929, with continuing support for the institute's operations under Ernst Rüdin over the next several years. Funding an experiment conducted by Vanderbilt University where they gave 800 pregnant women radioactive iron, 751 of which were pills, without their consent.
In a 1969 article published in the American Journal of Epidemiology, it was estimated that three children had died from the experiment. As of 2015, the Foundation was ranked as the 39th largest U. S. foundation by total giving. By year-end 2016 assets were tallied with annual grants of $173 million. On January 5, 2017, the board of trustees announced the unanimous selection of Dr. Rajiv Shah to serve as the 13th president of the foundation. Shah became the youngest person, at 43, first-ever Indian-American to serve as president of the foundation, he assumed the position March 1, succeeding Judith Rodin who served as president for nearly twelve years and announced her retirement, at age 71, in June 2016. Rodin in turn had succeeded Gordon Conway in 2005. A former president of the University of Pennsylvania, Rodin was the first woman to head the foundation. Rockefeller's interest in philanthropy and Public Relations began in 1904, influenced by Ida Tarbell's book published about Standard Oil crimes, The History of the Standard Oil Company, which prompted him to whitewash the Rockefeller image.
His initial idea to set up a large-scale foundation occurred in 1901, but it was not until 1906 that Senior's famous business and philanthropic advisor, Frederick Taylor Gates revived the idea, saying that Rockefeller's fortune was rolling up so fast his heirs would "dissipate their inheritances or become intoxicated with power", unless he set up "permanent corporate philanthropies for the good of Mankind". It was in 1906 that the Russell Sage Foundation was established, though its program was limited to working women and social ills. Rockefeller's would thus not be the first foundation in America, but it brought to it unprecedented international scale and scope. In 1909 he signed over 73,000 shares of Standard Oil of New Jersey, valued at $50 million, to the three inaugural trustees, Junior and Harold Fowler McCormick, the first installment of a projected $100 million endowment, they applied for a federal charter for the foundation in the US Senate in 1910, with at one stage Junior secretly meeting with President William Howard Taft, through the aegis of Senator Nelson Aldrich, to hammer out concessions.
However, because of the ongoing antitrust suit against Standard Oil at the time, along with deep suspicion in some quarters of undue Rockefeller influence on the spending of the endowment, the end result was that Senior and Gates withdrew the bill from Congress in order to seek a state charter. On May 14, 1913, New York Governor William Sulzer approved a state charter for the foundation – two years after the Carnegie Corporation – with Junior becoming the first president. With its large-scale endowment, a large part of Senior's fortune was insulated from inheritance taxes; the total benefactions of both him and Junior and their philanthropies in the end would far surpass Carnegie's endowments, his biographer Ron Chernow states, ranking Rockefeller as "the greatest philanthropist in American history." The first secretary of the foundation was Jerome Davis Greene, the former secretary of Harvard University, who wrote a "memorandum on principles and policies" for an early meeting of the trustees that established a rough framework for the foundation's work.
On December 5, the Board made its first grant of $100,000 to the American Red Cross to purchase property for its headquarters in Washington, D. C. At the beginning the foundation was global in its approach and concentrated in its first decade on the sciences, public health and medical education, it was located within the family office at Standard Oil's headquarters at 26 Broadway shifting to the GE Building, along with the newly named family office, Room 5600, at Rockefeller Center. In 1913 the foundation set up the International Health Commission, the first appropriation of funds for work outside the US, which launched the foundation into international public heal
Goldsmith
A goldsmith is a metalworker who specializes in working with gold and other precious metals. Goldsmiths have made silverware, goblets and serviceable utensils, ceremonial or religious items, using Kintsugi, but the rising prices of precious metals have curtailed the making of such items to a large degree. Goldsmiths must be skilled in forming metal through filing, sawing, forging and polishing metal; the trade has often included jewellery-making skills, as well as the similar skills of the silversmith. Traditionally, these skills had been passed along through apprenticeships, more jewellery arts schools specializing in teaching goldsmithing and a multitude of skills falling under the jewellery arts umbrella are available. Many universities and junior colleges offer goldsmithing and metal arts fabrication as a part of their fine arts curriculum. At least in Europe, goldsmiths performed many of the functions now regarded as part of banking providing custody of valuable items and currency exchange, though they were restrained from lending at interest, regarded as usury.
Compared to other metals, gold is malleable, rare, it is the only solid metallic element with a yellow color. It may be melted and cast without the problems of oxides and gas that are problematic with other metals such as bronzes, for example, it is easy to "pressure weld", wherein to clay two small pieces may be pounded together to make one larger piece. Gold is classified as a noble metal --, it is found in its native form, lasting indefinitely without oxidization and tarnishing. Gold has been worked by humans in all cultures where the metal is available, either indigenously or imported, the history of these activities is extensive. Superbly made objects from the ancient cultures of Africa, Europe, North America and South America grace museums and collections throughout the world; some pieces date back thousands of years and were made using many techniques that still are used by modern goldsmiths. Techniques developed by some of those goldsmiths achieved a skill level, lost and remained beyond the skills of those who followed to modern times.
Researchers attempting to uncover the chemical techniques used by ancient artisans have remarked that their findings confirm that "the high level of competence reached by the artists and craftsmen of these ancient periods who produced objects of an artistic quality that could not be bettered in ancient times and has not yet been reached in modern ones."In medieval Europe goldsmiths were organized into guilds and were one of the most important and wealthiest of the guilds in a city. The guild kept records of the marks they used on their products; these records, when they survive, are useful to historians. Goldsmiths acted as bankers, since they dealt in gold and had sufficient security for the safe storage of valuable items. In the Middle Ages, goldsmithing included silversmithing as well, but the brass workers and workers in other base metals were members of a separate guild, since the trades were not allowed to overlap. Many jewelers were goldsmiths; the Sunar caste is one of the oldest communities in goldsmithing in India, whose superb gold artworks were displayed at The Great Exhibition of 1851 in London.
In India,'Vishwakarma' are the goldsmith caste. The printmaking technique of engraving developed among goldsmiths in Germany around 1430, who had long used the technique on their metal pieces; the notable engravers of the fifteenth century were either goldsmiths, such as Master E. S. or the sons of goldsmiths, such as Martin Schongauer and Albrecht Dürer. A goldsmith might have a wide array of skills and knowledge at their disposal. Gold, being the most malleable metal of all, offers unique opportunities for the worker. In today's world a wide variety of other metals platinum alloys may be used frequently. 24 karat is pure gold and was known as fine gold. Because it is so soft, however, 24 karat gold is used, it is alloyed to make it stronger and to create different colors. The gold may be cast into some item usually with the lost wax casting process, or it may be used to fabricate the work directly in metal. In the latter case, the goldsmith will use a variety of tools and machinery, including the rolling mill, the drawplate, swage blocks and other forming tools to make the metal into shapes needed to build the intended piece.
Parts are fabricated through a wide variety of processes and assembled by soldering. It is a testament to the history and evolution of the trade that those skills have reached an high level of attainment and skill over time. A fine goldsmith can and will work to a tolerance approaching that of precision machinery, but using only his eyes and hand tools. Quite the goldsmith's job involves the making of mountings for gemstones, in which case they are referred to as jewelers.'Jeweller', however, is a term reserved for a person who deals in jewellery and not to be confused with a goldsmith, gemologist, diamond cutter, diamond setters. A'jobbing jeweller' is the term for a jeweller who undertakes a small basic amount of jewellery repair and alteration. Paul de Lamerie Paul Storr Lorenzo Ghiberti Benvenuto Cellini Johannes Gutenberg House of Fabergé Jean-Valentin Morel Adrien Vachette Gaspard van der Heyden Jocelyn Burton Lois Etherington Betteridge Andrea Cagnetti - Akelo William Claude Harper Mary Lee Hu Linda M
Trygve Haavelmo
Trygve Magnus Haavelmo, born in Skedsmo, was an economist whose research interests centered on econometrics. He received the Nobel Memorial Prize in Economic Sciences in 1989. After attending Oslo Cathedral School, Haavelmo received a degree in economics from the University of Oslo in 1930 and joined the Institute of Economics with the recommendation of Ragnar Frisch. Haavelmo was Frisch’s assistant for a period of time until he was appointed as head of computations for the institute. In 1936, Haavelmo studied statistics at University College London while he subsequently traveled to Berlin and Oxford for additional studies. Haavelmo assumed a lecturing position at the University of Aarhus in 1938 for one year and in the subsequent year was offered an academic scholarship to travel abroad and study in the United States. During World War II he worked with Nortraship in the Statistical Department in New York City, he received his Ph. D. in 1946 for his work on The Probability Approach in Econometrics.
He was a Professor of economics and statistics at the University of Oslo between 1948–79 and was the trade department head of division from 1947–48. Haavelmo acquired a prominent position in modern economics through his logical critique of a series of custom conceptions in mathematical analysis. In 1989, Haavelmo was awarded the Nobel Prize in Economics "for his clarification of the probability theory foundations of econometrics and his analyses of simultaneous economic structures."Haavelmo resided at Østerås. He died on 28 July 1999 in Oslo. Judea Pearl wrote "Haavelmo was the first to recognize the capacity of economic models to guide policies" and "presented a mathematical procedure that takes an arbitrary model and produces quantitative answers to policy questions". According to Pearl, "Haavelmo's paper, ‘The Statistical Implications of a System of Simultaneous Equations’, marks a pivotal turning point, not in the statistical implications of econometric models, as historians presume, but in their causal counterparts."
Haavelmo's idea that an economic model depicts a series of hypothetical experiments and that policies can be simulated by modifying equations in the model became the basis of all used formalisms of econometric causal inference. It was first operationalized by Robert H. Strotz and Herman Wold who advocated "wiping out" selected equations, translated into graphical models as "wiping out" incoming arrows; this operation has subsequently led to Pearl's "do"-calculus and to a mathematical theory of counterfactuals in econometric models. Pearl further speculates that the reason economists do not appreciate these revolutionary contributions of Haavelmo is because economists themselves have still not reached consensus of what an economic model stands for, as attested by profound disagreements among econometric textbooks. List of publications nobelprize.org bio Nobel Prize Lecture. at the Wayback Machine Model Discovery and Trygve Haavelmo’s Legacy by David F. Hendry and Søren Johansen.] Trygve Haavelmo Growth Model by Elmer G. Wiens Trygve Haavelmo.
The Concise Encyclopedia of Economics. Library of Economics and Liberty. Liberty Fund. 2008. Pearl, Judea. "Trygve Haavelmo and the Emergence of Causal Calculus". Forthcoming, Econometric Theory, special issue on Haavelmo Centennial. UCLA Computer Science Department, Technical Report R-391. Chen, Bryant. "Regression and Causation: A Critical Examination of Six Econometrics Textbooks". Real-World Economics Review. 65: 2–20
University of Oslo Faculty of Law
The Faculty of Law of the University of Oslo is Norway's oldest law faculty, established in 1811 as one of the four original faculties of The Royal Frederick University. Alongside the law faculties in Copenhagen and Uppsala, it is one of Scandinavia's leading institutions of legal education and research. Prior to 1811, the University of Copenhagen was the only university of Denmark-Norway, the curriculum of the new law faculty in Christiania was based on that of the University of Copenhagen Faculty of Law and long retained strong similarities after the dissolution of the Dano-Norwegian union in 1814; as the only faculty of law in Norway until 1980, it traditionally educated all lawyers of Norway and remains the country's most important law faculty, educating around 75% of all new legal candidates in Norway. Its law programme is one of the most competitive programmes to get into at any Norwegian university, with an acceptance rate of 12%; the faculty offers education and conducts research in both law and in related areas such as criminology and sociology of law, also in economics.
The faculty occupies the old university campus in the centre of Oslo, near the National Theatre, the Royal Palace, the Parliament, constructed 1841–1851 by Christian Heinrich Grosch with the assistance of world-famous Prussian architect Karl Friedrich Schinkel in Schinkel's neoclassical style, with strong similarities to Schinkel's famous museums on the Museum Island in Berlin. The old campus includes three main buildings, called Domus Academica, Domus Media and Domus Bibliotheca, centered on the University Square and facing Karl Johans gate; the Nobel Peace Prize was awarded in the atrium of the central building of the old campus, Domus Media, 1947–1989, since 2003, the Abel Prize is awarded in this building. The Parliament of Norway convened in the Old Ceremonial Hall in Domus Academica 1854–1866; the faculty publishes several academic journals, including the English-language journal Oslo Law Review. The University of Copenhagen was founded in 1479; as there was no university in Norway itself, the University of Copenhagen served both Denmark and Norway during the countries' personal union, the University of Copenhagen had both Norwegian students and teachers.
With the rise of absolute monarchy and a more professional civil service, legal education became of central importance by the early 18th century. During the Napoleonic Wars, after years of discussion, The Royal Frederick University in Norway was established in 1811 and named in honour of Frederick VI of Denmark and Norway, the Faculty of Law was one of the four original faculties, ranking second after the Faculty of Theology and before the Faculty of Medicine and the Faculty of Arts. In 1816, its first lecturers were appointed by the government: Lorents Lange was appointed Professor of Jurisprudence, Henrik Lauritz Nicolai Steenbuch was appointed Lecturer in Jurisprudence; as a sovereign kingdom, Norway always had its own laws, but in 1687, Norway received its Norwegian Code, nearly identical to the Danish Code, parts of which remain in effect until this day. Norway and Denmark hence in fact many of the same laws; the curriculum of the Faculty of Law in Christiania was hence to a large degree a direct continuation of the curriculum and traditions of the University of Copenhagen.
Similarities exist until this day, although they have been weakened. The field of economics as an academic discipline in Norway evolved at the Faculty of Law. In 1840, a chair in "Jurisprudence and Statistics" was created by the King; the most important programme of the Faculty of Law is the 5-year legal education leading to a Master of Laws degree, known in Norwegian as master i rettsvitenskap, a protected title under Norwegian law. The programme replaced the former six-year programme leading to a Candidate of Law degree, created in 1736 at the University of Copenhagen and retained at the Royal Frederick University from 1811; the Master or Laws or the former Candidate of Law are the only degrees qualifying for legal work in Norway. The graduates have a monopoly on a number of occupations, such as advocate, judge and, all the high ranks of the Norwegian Police Service and a number of senior civil servant positions. Norway has a united legal profession and all persons working in legal occupations have the same education.
Alongside the programme in medicine, the programme in law in Oslo is one of the most competitive to get into at any Norwegian university with an acceptance rate of 12%. Although students do not receive a formal degree before they have completed the five-year programme, the first four years correspond to an American J. D. degree. In the fifth year, students write a thesis corresponding to one semester and take advanced courses of their choice corresponding to one semester. Alternatively, they may choose to write corresponding to a full year. Parts of the fifth year, or the full year, may be taken abroad; the fifth year leads to the Master of Laws degree. From 1840 to 1966, the field of economics was part of the Faculty of Law, most of the professors of economics until the mid 20th century had a background in law. Prior to 1966, the Faculty of Law conferred the degree cand.oecon. Created in 1905, a 2-year supplementary degree in economics intended for
.jpg)
