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Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charlottenburg
Charlottenburg () is a Boroughs and localities of Berlin, locality of Berlin within the borough of Charlottenburg-Wilmersdorf. Established as a German town law, town in 1705 and named after Sophia Charlotte of Hanover, Queen consort of Kingdom of Prussia, Prussia, it is best known for Charlottenburg Palace, the largest surviving royal palace in Berlin, and the adjacent museums. Charlottenburg was an independent city to the west of Berlin until 1920 when it was incorporated into "Greater Berlin Act, Groß-Berlin" (Greater Berlin) and transformed into a borough. In the course of Berlin's 2001 administrative reform it was merged with the former borough of Wilmersdorf becoming a part of a new borough called Charlottenburg-Wilmersdorf. Later, in 2004, the new borough's districts were rearranged, dividing the former borough of Charlottenburg into the localities of Charlottenburg proper, Westend (Berlin), Westend and Charlottenburg-Nord. Geography Charlottenburg is located in Berlin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Method
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac and u'' \equiv \frac. in the vicinity of the regular singular point z=0. One can divide by z^2 to obtain a differential equation of the form u'' + \fracu' + \fracu = 0 which will not be solvable with regular power series methods if either or are not analytic at . The Frobenius method enables one to create a power series solution to such a differential equation, provided that ''p''(''z'') and ''q''(''z'') are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). Explanation The method of Frobenius is to seek a power series solution of the form u(z)=z^r \sum_^\infty A_k z^k, \qquad (A_0 \neq 0) Differentiating: u'(z)=\sum_^\infty (k+r)A_kz^ u''(z)=\sum_^\infty (k+r-1)(k+r)A_kz^ Substit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zürich
Zürich () is the list of cities in Switzerland, largest city in Switzerland and the capital of the canton of Zürich. It is located in north-central Switzerland, at the northwestern tip of Lake Zürich. As of January 2020, the municipality has 434,335 inhabitants, the Urban agglomeration, urban area 1.315 million (2009), and the Zürich metropolitan area 1.83 million (2011). Zürich is a hub for railways, roads, and air traffic. Both Zurich Airport and Zürich Hauptbahnhof, Zürich's main railway station are the largest and busiest in the country. Permanently settled for over 2,000 years, Zürich was founded by the Roman Empire, Romans, who called it '. However, early settlements have been found dating back more than 6,400 years (although this only indicates human presence in the area and not the presence of a town that early). During the Middle Ages, Zürich gained the independent and privileged status of imperial immediacy and, in 1519, became a primary centre of the Protestant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Eduard Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Life Kummer was born in Sorau, Brandenburg (then part of Prussia). He was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay (''De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis''), which was eventually published a year later. In 1840, Kummer married Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Felix Mendelssohn and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet. His second wife (whom he married soon after the death of Ottilie in 1848), Bertha Cauer, was a maternal cous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886". Weber, Heinrich L. 1891–1892Kronecker 2:5-23. (The quote is on p. 19.) Kronecker was a student and lifelong friend of . Biography Leopold Kronecker was born ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Protestant
Protestantism is a Christian denomination, branch of Christianity that follows the theological tenets of the Reformation, Protestant Reformation, a movement that began seeking to reform the Catholic Church from within in the 16th century against what its followers perceived to be growing Criticism of the Catholic Church, errors, abuses, and discrepancies within it. Protestantism emphasizes the Christian believer's justification by God in faith alone (') rather than by a combination of faith with good works as in Catholicism; the teaching that Salvation in Christianity, salvation comes by Grace in Christianity, divine grace or "unmerited favor" only ('); the Universal priesthood, priesthood of all faithful believers in the Church; and the ''sola scriptura'' ("scripture alone") that posits the Bible as the sole infallible source of authority for Christian faith and practice. Most Protestants, with the exception of Anglo-Papalism, reject the Catholic doctrine of papal supremacy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Manifold
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin,B. Dubrovin: ''Geometry of 2D topological field theories.'' In: Springer LNM, 1620 (1996), pp. 120–348. is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology. The broadest definition is in the category of Riemannian supermanifolds. We will limit the discussion here to smooth (real) manifolds. A restriction to complex manifolds is also possible. Definition Let ''M'' be a smooth manifold. An ''affine flat'' structure on ''M'' is a sheaf ''T''''f'' of vector spaces that pointwisely span ''TM'' the tangent bundle and the tangent bracket of pairs of its sections vanishes. As a local example consider the coordinate vectorfields over a chart of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...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] |
Differential Equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''. Period lattice and fundamental domain Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
