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Wirtinger Inequality (2-forms)
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups, and knot theory. Biography He was born at Ybbs on the Danube and studied at the University of Vienna, where he received his doctorate in 1887, and his habilitation in 1890. Wirtinger was greatly influenced by Felix Klein with whom he studied at the University of Berlin and the University of Göttingen. Honours In 1907 the Royal Society of London awarded him the Sylvester Medal, for his contributions to the general theory of functions. Work Research activity He worked in many areas of mathematics, publishing 71 works. His first significant work, published in 1896, was on theta functions. He proposed as a generalization of eigenvalues, the concept of the spectrum of an operator, in an 1897 paper; the concept was further extended by David Hilbert and now it forms the main object of investigation in the field of spectral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ybbs An Der Donau
Ybbs an der Donau (, ; short: Ybbs; Central Bavarian: ''Ybbs aun da Donau'') is a town in Austria. It was established in 1317. Throughout the town, from the intersection of the important trade routes and along the Danube the town has preserved a site that already had great economic importance during the Middle Ages. Toponymy The valley of the Ybbs river is called: ''Ybbs Valley'', or ''Ybbs Field'' (). History In 788, Ybbs Field () was the site of a battle, between Franks and Avars. Railroad bridge was dive-bombed by 14th FG on 26 March 1945 at 1020 hrs; Direct hit on abutments, south approach cut, main line blocked. Coat of arms On a silver shield lies a red city wall with battlements that an open gate and raised portcullis, which are dominated by two towers. Between the towers floats a green Linden bough, and the red-white-red ''Bindenschild''. Colors: Red-White-Red Coat of Arms Bestowal: unknown; at least since the 14th century. International relations Twin towns — Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wirtinger Derivatives
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. Historical notes Early days (1899–1911): the work of Henri Poincaré Wirtinger derivatives were used in complex analysis at least as early as in the paper , as briefly noted by and by . In the third paragraph of his 1899 paper, Henri Poincaré first define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...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, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); 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's theorem. The number of known mathematicians grew when Pythagoras of Samos () 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 math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austria
Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Austria, most populous city and state. Austria is bordered by Germany to the northwest, the Czech Republic to the north, Slovakia to the northeast, Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the west. The country occupies an area of and has Austrians, a population of around 9 million. The area of today's Austria has been inhabited since at least the Paleolithic, Paleolithic period. Around 400 BC, it was inhabited by the Celts and then annexed by the Roman Empire, Romans in the late 1st century BC. Christianization in the region began in the 4th and 5th centuries, during the late Western Roman Empire, Roman period, followed by the arrival of numerous Germanic tribes during the Migration Period. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Medal
The Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian chair of geometry, Savilian Professor of Geometry at the University of Oxford in the 1880s, and first awarded in 1901, having been suggested by a group of Sylvester's friends (primarily Raphael Meldola) after his death in 1897. Initially awarded every three years with a prize of around £900, the Royal Society have announced that starting in 2009 it will be awarded every two years instead, and is to be aimed at 'early to mid career stage scientist' rather than an established mathematician. The award winner is chosen by the Society's A-side awards committee, which handles physical rather than biological science awards. , 45 medals have been awarded, of which all but 10 have been awarded to citizens of the United Kingdom, two to citizens of France and United States, and on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wirtinger's Representation And Projection Theorem
In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace \left.\right. H_2 of the simple, unweighted holomorphic Hilbert space \left.\right. L^2 of functions square-integrable over the surface of the unit disc \left.\right.\ of the complex plane, along with a form of the orthogonal projection from \left.\right. L^2 to \left.\right. H_2 . Wirtinger's paper contains the following theorem presented also in Joseph L. Walsh's well-known monograph (p. 150) with a different proof. ''If'' \left.\right.\left. F(z)\right. ''is of the class'' \left.\right. L^2 on \left.\right. , z, <1 , ''i.e.'' : ''where is the |
