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Nother Look At Bach's Fugue In E-flat
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile aphysikerinnen.de/noetherlebenslauf.html). Sometimes ''Emmy'' is mistakenly reported as a short form for ''Amalie'', or misreported as "Emily". e.g. (, ; ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's First and Second Theorem, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she devel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erlangen
Erlangen (; East Franconian German, East Franconian: ''Erlang'', Bavarian language, Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative district Erlangen), and with 116,062 inhabitants (as of 30 March 2022), it is the smallest of the eight major cities (''Town#Germany, Großstadt'') in Bavaria. The number of inhabitants exceeded the threshold of 100,000 in 1974, making Erlangen a major city according to the statistical definition officially used in Germany. Together with Nuremberg, Fürth, and Schwabach, Erlangen forms one of the three metropolises in Bavaria. With the surrounding area, these cities form the Nuremberg Metropolitan Region, European Metropolitan Region of Nuremberg, one of 11 metropolitan areas in Germany. The cities of Nuremberg, Fürth, and Erlangen also form a triangle on a map, which represents the heartland of the Nuremberg conurbation. An element of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the family to China to work as missionaries, and he did not return to Europe until he was nine. After his schooling, Witt went to the University of Freiburg and the University of Göttingen. He joined the NSDAP (Nazi Party) and was an active party member. Witt was awarded a Ph.D. at the University of Göttingen in 1934 with a thesis titled: "Riemann-Roch theorem and zeta-Function in hypercomplexes" (Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen) that was supervised by Gustav Herglotz with Emmy Noether suggesting the top for the doctorate. He qualified to become a lecturer and gave guest lectures in Göttingen and Hamburg. He became associated with the team led by Helmut Hasse who led his habilitation. In June 1936 gave his habil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether's Second Theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. :Translated in The action ''S'' of a physical system is an integral of a so-called Lagrangian function ''L'', from which the system's behavior can be determined by the principle of least action. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by ''k'' arbitrary functions and their derivatives up to order ''m'', then the functional derivatives of ''L'' satisfy a system of ''k'' differential equations. Noether's second theorem is sometimes used in gauge theory. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model. The theorem is named after Emmy Noether. See also * Noether's first theorem * Noether identities * Gauge symmetry (mathematics) In mathematics, any Lagrangian sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germans
, native_name_lang = de , region1 = , pop1 = 72,650,269 , region2 = , pop2 = 534,000 , region3 = , pop3 = 157,000 3,322,405 , region4 = , pop4 = 21,000 3,000,000 , region5 = , pop5 = 125,000 982,226 , region6 = , pop6 = 900,000 , region7 = , pop7 = 142,000 840,000 , region8 = , pop8 = 9,000 500,000 , region9 = , pop9 = 357,000 , region10 = , pop10 = 310,000 , region11 = , pop11 = 36,000 250,000 , region12 = , pop12 = 25,000 200,000 , region13 = , pop13 = 233,000 , region14 = , pop14 = 211,000 , region15 = , pop15 = 203,000 , region16 = , pop16 = 201,000 , region17 = , pop17 = 101,000 148,00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rufname
Personal names in German-speaking Europe consist of one or several given names (''Vorname'', plural ''Vornamen'') and a surname (''Nachname, Familienname''). The ''Vorname'' is usually gender-specific. A name is usually cited in the " Western order" of "given name, surname", unless it occurs in an alphabetized list of surnames, e.g. " Bach, Johann Sebastian". In this, the German conventions parallel the naming conventions in most of Western and Central Europe, including English, Dutch, Italian, and French. There are some vestiges of a patronymic system as they survive in parts of Eastern Europe and Scandinavia, but these do not form part of the official name. Women traditionally adopted their husband's name upon marriage and would occasionally retain their maiden name by hyphenation, in a so-called '' Doppelname'', e.g. "Else Lasker-Schüler". Recent legislation motivated by gender equality now allows a married couple to choose the surname they want to use, including an option fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmy (given Name)
Emmy is a feminine (sometimes also masculine) given name. Orthographic variants include Emme, Emmi and Emmie. The name is in many instances a hypocoristic of either Emma (itself being in origin a hypocoristic of a number of ancient Germanic names beginning in ''Ermen-'') or Emily, or Emmanuel (Emmanuelle). It came to be used as a separate (rare) German name, given officially in Germany from the later 19th century. As an officially given feminine name, Emmy ranked 66th in Sweden and 89th in France as of 2010. statistics cited aftebehindthename.com In France, rank 89 was reached after a steady rise in popularity during the 2000s, starting out at rank 281 in 2001. Emmy is rarely also encountered as a surname. Notable people with the name include: * Emmy Andriesse (1914–1953), Dutch photographer * Emmie Charayron (born 1990), French triathlete * Emmy Krüger (1886-1976), German operatic soprano * Emmy Loose (1914-1987), Austrian operatic soprano * Emmie te Nijenhuis (born 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lasker–Noether Theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Laskerâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
