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Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German-born American mathematician, known for work in mathematical analysis and number theory. Biography Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göttingen; Constantin Carathéodory supervised his dissertation. In 1919, he became '' privatdozent'' under Constantin Carathéodory at University of Berlin. In 1922, he became an assistant professor at the University of Hamburg, where he supervised budding mathematicians like Theodor Estermann. He was dismissed from his position at the University of Breslau by the Nazis in 1933 due to his public support of the Weimar Republic, and emigrated from Europe in 1934. After leaving Germany, he moved to Philadelphia and worked at the University of Pennsylvania until his retirement in 1962; he held the Thomas A. Scott Professorship of Mathematics at Pennsylvania from 1956 to 1962. Rademac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamburg-Wandsbek
Wandsbek () is an urban quarter in the Wandsbek borough of Hamburg, Germany, and the former city Wandsbek in the Duchy of Holstein. In 2020 the population was 36,671. History Wandsbek was once part of the county ''Stormarn''. Its villages were first mentioned in the middle of the 13th century. The name ''Wandsbek'', ''Wandsbeck'' or (older) ''Wantesbeke'' derives from old Low Saxon ("Low German") for "border river" and the river Wandse was a natural territorial border. An old Danish phrase for stating that something is a fraud / unreliable is to claim that ''"det gælder ad Wandsbek Vandsbæktil"'' (i.e. ''"this is valid in Wandsbeck."''). Wandsbek was one of the three locations in the Danish monarchy where the first lottery drew its numbers, and this expression dates from the early years of this lottery's life where a number of people tried to claim prizes in Copenhagen with tickets from Wandsbeck. Since each of the three towns drew its own set of numbers, a ticket from one tow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (number Theory)
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of ends in the digit 4 or 9, the number of partitions of will be divisible by 5. Definition and examples For a positive integer , is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Sum
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number of functional equations; this article lists only a small fraction of these. Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers. Definition Define the sawtooth function (\!( \, )\!) : \mathbb \rightarrow \mathbb as :(\!(x)\!)=\begin x-\lfloor x\rfloor - 1/2, &\mboxx\in\mathbb\setminus\mathbb;\\ 0,&\mboxx\in\mathbb. \end We then let :D: \mathbb^2\times (\mathbb-\)\to \mathbb be defined by :D(a,b;c)=\sum_ \left(\!\!\left( \frac \right)\!\!\right) \! \left(\!\!\left( \frac \right)\!\!\right), the terms on the right being the Dedekind su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genetics
Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar working in the 19th century in Brno, was the first to study genetics scientifically. Mendel studied "trait inheritance", patterns in the way traits are handed down from parents to offspring over time. He observed that organisms (pea plants) inherit traits by way of discrete "units of inheritance". This term, still used today, is a somewhat ambiguous definition of what is referred to as a gene. Trait inheritance and molecular inheritance mechanisms of genes are still primary principles of genetics in the 21st century, but modern genetics has expanded to study the function and behavior of genes. Gene structure and function, variation, and distribution are studied within the context of the cell, the organism (e.g. dominance), and within the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas A
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Illinois * Thomas, Indiana * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts, entertainment, and media * ''Thomas'' (Burton novel) 1969 nove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philadelphia
Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Since 1854, the city has been coextensive with Philadelphia County, the most populous county in Pennsylvania and the urban core of the Delaware Valley, the nation's seventh-largest and one of world's largest metropolitan regions, with 6.245 million residents . The city's population at the 2020 census was 1,603,797, and over 56 million people live within of Philadelphia. Philadelphia was founded in 1682 by William Penn, an English Quaker. The city served as capital of the Pennsylvania Colony during the British colonial era and went on to play a historic and vital role as the central meeting place for the nation's founding fathers whose plans and actions in Philadelphia ultimately inspired the American Revolution and the nation's inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weimar Republic
The Weimar Republic (german: link=no, Weimarer Republik ), officially named the German Reich, was the government of Germany from 1918 to 1933, during which it was a constitutional federal republic for the first time in history; hence it is also referred to, and unofficially proclaimed itself, as the German Republic (german: Deutsche Republik, link=no, label=none). The state's informal name is derived from the city of Weimar, which hosted the constituent assembly that established its government. In English, the republic was usually simply called "Germany", with "Weimar Republic" (a term introduced by Adolf Hitler in 1929) not commonly used until the 1930s. Following the devastation of the First World War (1914–1918), Germany was exhausted and sued for peace in desperate circumstances. Awareness of imminent defeat sparked a revolution, the abdication of Kaiser Wilhelm II, formal surrender to the Allies, and the proclamation of the Weimar Republic on 9 November 1918. In its i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nazi
Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right totalitarian political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Nazi Germany. During Hitler's rise to power in 1930s Europe, it was frequently referred to as Hitlerism (german: Hitlerfaschismus). The later related term " neo-Nazism" is applied to other far-right groups with similar ideas which formed after the Second World War. Nazism is a form of fascism, with disdain for liberal democracy and the parliamentary system. It incorporates a dictatorship, fervent antisemitism, anti-communism, scientific racism, and the use of eugenics into its creed. Its extreme nationalism originated in pan-Germanism and the ethno-nationalist '' Völkisch'' movement which had been a prominent aspect of German nationalism since the late 19th century, and it was strongly influenced by the paramilitary groups that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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