Frank Spitzer
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Frank Spitzer
Frank Ludvig Spitzer (July 24, 1926 – February 1, 1992) was an Austrian-born, Jewish-American mathematician who was a longtime professor at Cornell University and made fundamental contributions to probability theory, especially the theory of random walks, Brownian motion, and fluctuation theory, and then the theory of interacting particle systems. Other areas he made contributions to include percolation theory and the Wiener sausage. He focussed broadly on "phenomena", rather than any one of the many specific theorems that might help to articulate a given phenomenon. His book ''Principles of Random Walk'', first published in 1964, remains a well-cited classic. Spitzer was born on July 24, 1926, in Vienna, into an Austrian Jewish family. By the time he was twelve years old, the Nazi threat in Austria was evident. His parents were able to send him to a summer camp for Jewish children in Sweden, and, as a result, Spitzer spent all of the World War II years in Sweden. He lived w ...
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Vienna
Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. Its larger metropolitan area has a population of nearly 2.9 million, representing nearly one-third of the country's population. Vienna is the Culture of Austria, cultural, Economy of Austria, economic, and Politics of Austria, political center of the country, the List of cities in the European Union by population within city limits, fifth-largest city by population in the European Union, and the most-populous of the List of cities and towns on the river Danube, cities on the river Danube. The city lies on the eastern edge of the Vienna Woods (''Wienerwald''), the northeasternmost foothills of the Alps, that separate Vienna from the more western parts of Austria, at the transition to the Pannonian Basin. It sits on the Danube, and is ...
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Random Walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating random walk hypothesis, stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Realizations of random walks can be obtained by Monte Carlo Simulation, Monte Carlo simulation. Lattice random ...
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Stockholm
Stockholm (; ) is the Capital city, capital and List of urban areas in Sweden by population, most populous city of Sweden, as well as the List of urban areas in the Nordic countries, largest urban area in the Nordic countries. Approximately 1 million people live in the Stockholm Municipality, municipality, with 1.6 million in the Stockholm urban area, urban area, and 2.5 million in the Metropolitan Stockholm, metropolitan area. The city stretches across fourteen islands where Mälaren, Lake Mälaren flows into the Baltic Sea. Outside the city to the east, and along the coast, is the island chain of the Stockholm archipelago. The area has been settled since the Stone Age, in the 6th millennium BC, and was founded as a city in 1252 by Swedish statesman Birger Jarl. The city serves as the county seat of Stockholm County. Stockholm is the cultural, media, political, and economic centre of Sweden. The Stockholm region alone accounts for over a third of the country's Gros ...
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Royal Institute Of Technology
KTH Royal Institute of Technology (), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technology and is Sweden's largest technical university. Since 2018, KTH consists of five schools with four campuses in and around Stockholm. KTH was established in 1827 as the ''Teknologiska institutet'' (Institute of Technology) and had its roots in the ''Mekaniska skolan'' (School of Mechanics) that was established in 1798 in Stockholm. But the origin of KTH dates back to the predecessor of the ''Mekaniska skolan'', the ''Laboratorium mechanicum'', which was established in 1697 by the Swedish scientist and innovator Christopher Polhem. The ''Laboratorium mechanicum'' combined education technology, a laboratory, and an exhibition space for innovations. In 1877, KTH received its current name, ''Kungliga Tekniska högskolan'' (KTH Royal Institute of Technology). The Swedish king, His Majesty Carl XVI Gustaf, is the pa ...
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Swedish Language
Swedish ( ) is a North Germanic languages, North Germanic language from the Indo-European languages, Indo-European language family, spoken predominantly in Sweden and parts of Finland. It has at least 10 million native speakers, making it the Germanic_languages#Statistics, fourth most spoken Germanic language, and the first among its type in the Nordic countries overall. Swedish, like the other North Germanic languages, Nordic languages, is a descendant of Old Norse, the common language of the Germanic peoples living in Scandinavia during the Viking Age. It is largely mutually intelligible with Norwegian language, Norwegian and Danish language, Danish, although the degree of mutual intelligibility is dependent on the dialect and accent of the speaker. Standard Swedish, spoken by most Swedes, is the national language that evolved from the Central Swedish dialects in the 19th century, and was well established by the beginning of the 20th century. While distinct regional Variety ( ...
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Sweden
Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic country by both area and population, and is the List of European countries by area, fifth-largest country in Europe. Its capital and largest city is Stockholm. Sweden has a population of 10.6 million, and a low population density of ; 88% of Swedes reside in urban areas. They are mostly in the central and southern half of the country. Sweden's urban areas together cover 1.5% of its land area. Sweden has a diverse Climate of Sweden, climate owing to the length of the country, which ranges from 55th parallel north, 55°N to 69th parallel north, 69°N. Sweden has been inhabited since Prehistoric Sweden, prehistoric times around 12,000 BC. The inhabitants emerged as the Geats () and Swedes (tribe), Swedes (), who formed part of the sea-faring peopl ...
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Summer Camp
A summer camp, also known as a sleepaway camp or residential camp, is a supervised overnight program for children conducted during the summer vacation from school in many countries. Children and adolescents who attend summer residential camps are known as . They are generally offered overnight accommodations for one or two weeks out in an outdoor natural campsite setting. Day camps, by contrast, offer the same types of experience in the outdoors but children return home each evening. Summer school is a different experience that is usually offered by local schools for their students focused on remedial education to ensure students are prepared for the upcoming academic year or in the case of high school students, to retake failed state comprehensive exams necessary for graduation. Summer residential and day camps may include an academic component but it is not a requirement. The traditional view of a summer camp as a woodland, wooded place with hiking, canoeing, campfires, et ...
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Nazism
Nazism (), formally named National Socialism (NS; , ), is the far-right totalitarian socio-political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Germany. During Hitler's rise to power, it was frequently referred to as Hitler Fascism () and Hitlerism (). The term " neo-Nazism" is applied to other far-right groups with similar ideology, which formed after World War II, and after Nazi Germany collapsed. Nazism is a form of fascism, with disdain for liberal democracy and the parliamentary system. Its beliefs include support for dictatorship, fervent antisemitism, anti-communism, anti-Slavism, anti-Romani sentiment, scientific racism, white supremacy, Nordicism, social Darwinism, homophobia, ableism, and the use of eugenics. The ultranationalism of the Nazis originated in pan-Germanism and the ethno-nationalist '' Völkisch'' movement which had been a prominent aspect of German ultranationalism since the late 19th centu ...
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History Of The Jews In Austria
The history of the Jews in Austria starts after the exodus of Jews from Judea under Roman occupation. There have been Jews in Austria since the 3rd century CE. Over the course of many centuries, the political status of the community rose and fell many times: during certain periods, the Jewish community prospered and enjoyed political equality, and during other periods it suffered pogroms, deportations to concentration camps and mass murder, and further antisemitism. The Holocaust drastically reduced the Jewish community in Austria and only 8,140 Jews remained in Austria according to the 2001 census. As of 2020, Austria had a Jewish population of 10,300 and a total of 33,000 when including any Austrian with at least one Jewish grandparent. Antiquity Jews have been in Austria since at least the 3rd century CE. In 2008, a team of archeologists discovered a third-century CE amulet in the form of a gold scroll with the words of the Jewish prayer Shema Yisrael (Hear, O Israel! ...
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Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ...
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Phenomenon
A phenomenon ( phenomena), sometimes spelled phaenomenon, is an observable Event (philosophy), event. The term came into its modern Philosophy, philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried Wilhelm Leibniz in this part of his philosophy, in which phenomenon and noumenon serve as interrelated technical terms. Far predating this, the Ancient Greek Philosophy, ancient Greek Pyrrhonism, Pyrrhonist philosopher Sextus Empiricus also used ''phenomenon'' and ''noumenon'' as interrelated technical terms. Common usage In popular usage, a ''phenomenon'' often refers to an extraordinary, unusual or notable event. According to the ''Dictionary of Visual Discourse'':In ordinary language 'phenomenon/phenomena' refer to any occurrence worthy of note and investigation, typically an untoward or unusual event, person or fact that is of special significance or otherwise notable. ...
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Percolation Theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger Glossary of graph theory, connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology). Introduction A representative question (and the etymology, source of the name) is as follows. Assume that some liquid is poured on top of some porosity, porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is mathematical model, modelled mathematically as a Grid graph, three-dimensional network of graph (discrete mathematics), vertices, usually called "sites", in which the graph (dis ...
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