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Hans Hermes
Hans Hermes (; 12 February 1912 – 10 November 2003) was a German mathematician and logician, who made significant contributions to the foundations of mathematical logic. Hermes was born in Neunkirchen, Germany. Personal life From 1931, Hermes studied mathematics, physics, chemistry, biology and philosophy at the University of Freiburg. In 1937, he passed the state examination in Münster and was attending there in 1938 when the physicist Adolf Kratzer was present. After that, he went on a scholarship to the University of Göttingen and then became an assistant at the University of Bonn. During World War II, he was a soldier on the Channel Island of Jersey until 1943 and then on to the Chemical Physics Institute of the Navy in Kiel. At the end of the war, he moved to Toplitzsee, where he was tasked with working on new encryption methods. In 1947, he became a lecturer at the University of Bonn where he took his habilitation, his thesis called ''Analytical manifolds in Riemannia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neunkirchen, Saarland
Neunkirchen (; pfl, Neinkeije) is a town and a municipality in Saarland, Germany. It is the largest town in, and the seat of the Neunkirchen (German district), district of Neunkirchen. It is situated on the river Blies, approx. 20 km northeast of Saarbrücken. With about 50,000 inhabitants, Neunkirchen is Saarland's second largest city. Overview The name of the town derives from "An der neuen Kirche" meaning "by the new church" not from "nine churches" as one might be tempted to assume. In the past, Neunkirchen's economy has been shaped almost exclusively by coal and steel. With the decline of this industry sector, Neunkirchen's economy had to face drastic changes and underwent a significant shift towards the service and retail sector, although smaller industries still remain. History Early history The earliest settlements in the area can be dated back to 700 BC. The oldest part of the town is the village of Wiebelskirchen north of the town centre; its name has been recor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toplitzsee
Lake Toplitz (German: ''Toplitzsee'') is a lake situated in a dense mountain forest high up in the Austrian Alps, from Salzburg in western Austria. It is surrounded by cliffs and forests in the Salzkammergut lake district, within the Totes Gebirge (dead mountains). The Toplitzsee water contains no oxygen below a depth of 20 m. Fish can survive only in the top 18 m, as the water below 20 m is salty, although bacteria and worms that can live without oxygen have been found below 20 m. In 1943 and 1944, the shore of Lake Toplitz served as a Nazi naval testing station. Using copper diaphragms, scientists experimented with different explosives, detonating up to 4,000 kg charges at various depths. Over £100 million of counterfeit pound sterling notes were claimed to have been dumped in the lake after Operation Bernhard, which was never fully put into action. In 1959, investigators recovered £700 million of counterfeit notes from the lake, that Hitler had planne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz-Dieter Ebbinghaus
Heinz-Dieter Ebbinghaus (born 22 February 1939 in Hemer, Province of Westphalia) is a German mathematician and logician. He received his PhD in 1967 at the University of Münster under Hans Hermes and Dieter Rödding. Ebbinghaus has written various books on logic, set theory and model theory, including a seminal work on Ernst Zermelo. His book ''Einführung in die mathematische Logik'', joint work with Jörg Flum and Wolfgang Thomas, first appeared in 1978 and became a standard textbook of mathematical logic in the German-speaking area. It is currently in its sixth edition (). An English edition of ''Mathematical Logic'' was published in the Springer-Verlag Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow boo ... series in 1984 (), with a second editio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Completeness
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete. A related concept is that of Turing equivalence two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by a Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing machine. A universal Turin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues And Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gisbert Hasenjaeger
Gisbert F. R. Hasenjaeger (June 1, 1919 – September 2, 2006) was a German mathematical logician. Independently and simultaneously with Leon Henkin in 1949, he developed a new proof of the completeness theorem of Kurt Gödel for predicate logic. He worked as an assistant to Heinrich Scholz at Section IVa of Oberkommando der Wehrmacht Chiffrierabteilung, and was responsible for the security of the Enigma machine. Personal life Gisbert Hasenjaeger went to high school in Mülheim, where his father was a lawyer and local politician. After completing school in 1936, Gisbert volunteered for labor service. He was drafted for military service in World War II, and fought as an artillerist in the Russian campaign, where he was badly wounded in January 1942. After his recovery, in October 1942, Heinrich Scholz got him employment in the Cipher Department of the High Command of the Wehrmacht (OKW/Chi), where he was the youngest member at 24. He attended a cryptography training course by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relationship with predictability. Perfect predictability implies strict determinism, but lack of predictability does not necessarily imply lack of determinism. Limitations on predictability could be caused by factors such as a lack of information or excessive complexity. In experimental physics, there are always observational errors determining variables such as positions and velocities. So perfect prediction is ''practically'' impossible. Moreover, in modern quantum mechanics, Werner Heisenberg's indeterminacy principle puts limits on the accuracy with which such quantities can be known. So such perfect predictability is also ''theoretically'' impossible. Laplace's demon Laplace's demon is a supreme intelligence who could completely predict the one possible future given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
