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Friedrich Schottky
Friedrich Hermann Schottky (24 July 1851 – 12 August 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and introduced Schottky groups and Schottky's theorem. Biography Friedrich Hermann Schottky was born in Breslau, Germany (now Wrocław, Poland). His father, Dr. Hermann Friedrich Schottky, was an English teacher and his mother, Louise Winkler, was a florist. He attended from 1860 to 1870, where his classmates included , , and Eberhard Gothein.From 1870 to 1874 he attended the University of Breslau. In 1875 Schottky received his doctorate, studying under Karl Weierstrass and Hermann von Helmholtz at Friedrich Wilhelm University of Berlin. Schottky was a lecturer at the University of Breslau from 1878 to 1882, a professor at the University of Zurich from 1882 to 1892, and a professor at Philipps University of Marburg from 1892 to 1902. In 1902, through his friendship with Ferdinand Georg Frobenius, Schottky was able to obtain a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Schnee
Walter Schnee (8 August 1885 – 10 June 1958) was a German mathematician. Biography Schnee was born on 8 August 1885 in Rawicz. From 1904 to 1908, he studied mathematics in Berlin. From 1909 to 1917, he worked at the University of Breslau. He then went to the University of Leipzig, where he stayed until 1954. He worked in the field of number theory. He died on 10 June 1958 in Leipzig Leipzig (, ; ; Upper Saxon: ; ) is the most populous city in the States of Germany, German state of Saxony. The city has a population of 628,718 inhabitants as of 2023. It is the List of cities in Germany by population, eighth-largest city in Ge .... References 1885 births 1958 deaths People from Rawicz People from the Province of Posen 20th-century German mathematicians {{Germany-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky's Theorem
In mathematical complex analysis, Schottky's theorem, introduced by is a quantitative version of Picard's theorem. It states that for a holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... ''f'' in the open unit disk that does not take the values 0 or 1, the value of , ''f''(''z''), can be bounded in terms of ''z'' and ''f''(0). Schottky's original theorem did not give an explicit bound for ''f''. gave some weak explicit bounds. gave a strong explicit bound, showing that if ''f'' is holomorphic in the open unit disk and does not take the values 0 or 1, then :\log , f(z), \le \frac(7+\max(0,\log , f(0), )). Several authors, such as , have given variations of Ahlfors's bound with better constants: in particular gave some bounds whose constants are in some sens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be holomorphically embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for 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 doubly periodic functions. Period lattice and fundamental domain If f is an elliptic function with periods \omega_1,\omega_2 it also holds ... [...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] |
Schottky Theorem
In mathematical complex analysis, Schottky's theorem, introduced by is a quantitative version of Picard's theorem. It states that for a holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... ''f'' in the open unit disk that does not take the values 0 or 1, the value of , ''f''(''z''), can be bounded in terms of ''z'' and ''f''(0). Schottky's original theorem did not give an explicit bound for ''f''. gave some weak explicit bounds. gave a strong explicit bound, showing that if ''f'' is holomorphic in the open unit disk and does not take the values 0 or 1, then :\log , f(z), \le \frac(7+\max(0,\log , f(0), )). Several authors, such as , have given variations of Ahlfors's bound with better constants: in particular gave some bounds whose constants are in some sens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Problem
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties. Geometric formulation More precisely, one should consider algebraic curves C of a given genus g, and their Jacobians \operatorname(C). There is a moduli space \mathcal_g of such curves, and a moduli space of abelian varieties, \mathcal_g, of dimension g, which are ''principally polarized''. There is a morphism\operatorname: \mathcal_g \to \mathcal_gwhich on points (geometric points, to be more accurate) takes isomorphism class /math> to operatorname(C)/math>. The content of Torelli's theorem is that \operatorname is injective (again, on points). The Schottky problem asks for a description of the image of \operatorname, denoted \mathcal_g = \operatorname(\mathcal_g). The dimension of \mathcal_g is 3g - 3, for g \geq 2, while the dimension of ''\mathcal_g'' is ''g''(''g'' + 1)/2. This m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by . Definition Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call the piece containing ''p'' the "exterior" of the curve, and the other piece its "interior". Suppose there are 2''g'' disjoint Jordan curves ''A''1, ''B''1,..., ''A''''g'', ''B''''g'' in the Riemann sphere with disjoint interiors. If there are Möbius transformations ''T''''i'' taking the outside of ''A''''i'' onto the inside of ''B''''i'', then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this. Properties By work of , a finitely generated Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic. A fundamental domain for the action of a Schottky group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Form
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form ''J'' of degree 4 and weight 8, introduced by as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corresponding to 4-dimensional abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ... that are the Jacobian varieties of genus 4 curves). showed that it is a multiple of the difference θ4(''E''8 ⊕ ''E''8) − θ4(''E''16) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. showed that it generates the 1-dimensional space of level 1 genus 4 weight 8 Siegel cusp forms. Ikeda showed that the Schottky form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
