field_of_work oskar bolza
Karl Theodor Wilhelm Weierstrass ( 31 October 1815  19 February 1897 ) was a German ___MASK94___ often cited as the " father of modern analysis ". Despite leaving university without a degree , he studied ___MASK65___ and trained as a teacher , eventually ___MASK70___ ___MASK65___ , ___MASK80___ , botany and gymnastics . Christian Felix Klein ( 25 April 1849  22 June 1925 ) was a German ___MASK94___ and ___MASK65___ educator , known for his work in ___MASK14___ , ___MASK53___ , non - Euclidean ___MASK28___ , and on the connections between ___MASK28___ and ___MASK14___ . His 1872 Erlangen Program , classifying geometries by their underlying symmetry groups , was a hugely influential synthesis of much of the ___MASK65___ of the day . In ___MASK65___ , a real number is a value that represents a quantity along a line . The adjective " real " in this context was introduced in the 17th century by René Descartes , who distinguished between real and imaginary roots of polynomials . ___MASK28___ ( from the ; " geo -" " earth ", "- metron " " measurement ") is a branch of mathematics concerned with questions of shape , size , relative position of figures , and the properties of space . A mathematician who works in the field of ___MASK28___ is called a geometer . Südliche Weinstraße is a district (" Kreis ") in the south of Rhineland - Palatinate , Germany . Neighboring districts are ( from west clockwise ) Südwestpfalz , Bad Dürkheim , the district - free city Neustadt ( Weinstraße ), Rhein - Pfalz - Kreis , Germersheim , and the ___MASK77___ " département " Bas - Rhin . The district - free city Landau is surrounded by the district . France , officially the ___MASK77___ Republic , is a country with territory in western Europe and several overseas regions and territories . The European , or metropolitan , area of France extends from the Mediterranean Sea to the English Channel and the North Sea , and from the Rhine to the Atlantic Ocean . Overseas France include ___MASK77___ Guiana on the South American continent and several island territories in the Atlantic , Pacific and Indian oceans . France spans and had a total population of almost 67 million people as of January 2017 . It is a unitary semi - presidential republic with the capital in Paris , the country ' s largest city and main cultural and commercial centre . Other major urban centres include Marseille , Lyon , Lille , Nice , Toulouse and Bordeaux . Bad Bergzabern is a municipality in the Südliche Weinstraße district , on the German Wine Route in Rhineland - Palatinate , Germany . It is situated near the border with France , on the south - eastern edge of the Palatinate forest , approximately southwest of Landau . ___MASK57___ is a field of ___MASK78___ that deals with maximizing or minimizing functionals , which are mappings from a set of functions to the real numbers . Functionals are often expressed as definite integrals involving functions and their derivatives . The interest is in " extremal " functions that make the functional attain a maximum or minimum value  or " stationary " functions  those where the rate of change of the functional is zero . In ___MASK65___ , non - Euclidean ___MASK28___ consists of two geometries based on axioms closely related to those specifying Euclidean ___MASK28___ . As Euclidean ___MASK28___ lies at the intersection of metric ___MASK28___ and affine ___MASK28___ , non - Euclidean ___MASK28___ arises when either the metric requirement is relaxed , or the parallel postulate is replaced with an alternative one . In the latter case one obtains hyperbolic ___MASK28___ and elliptic ___MASK28___ , the traditional non - Euclidean geometries . When the metric requirement is relaxed , then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non - Euclidean ___MASK28___ . Euclidean ___MASK28___ is a mathematical system attributed to the Alexandrian Greek mathematician Euclid , which he described in his textbook on ___MASK28___ : the " Elements ". Euclid ' s method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions ( theorems ) from these . Although many of Euclid ' s results had been stated by earlier mathematicians , Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system . The " Elements " begins with plane ___MASK28___ , still taught in secondary school as the first axiomatic system and the first examples of formal proof . It goes on to the solid ___MASK28___ of three dimensions . Much of the " Elements " states results of what are now called algebra and number theory , explained in geometrical language . ___MASK65___ ( from Greek  " máthma ",  knowledge , study , learning ) is the study of topics such as quantity ( numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of ___MASK65___ . In ___MASK65___ and abstract ___MASK68___ , group theory studies the algebraic structures known as groups . The concept of a group is central to abstract ___MASK68___ : other well - known algebraic structures , such as rings , fields , and vector spaces , can all be seen as groups endowed with additional operations and axioms . Groups recur throughout ___MASK65___ , and the methods of group theory have influenced many parts of ___MASK68___ . 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 . Oskar Bolza ( 12 May 1857 -- 5 July 1942 ) was a German ___MASK94___ , and student of Felix Klein . He was born in Bad Bergzabern , Rhenish Palatinate , known for his research in the ___MASK57___ , particularly influenced by Karl Weierstrass ' 1879 lectures on the subject . ___MASK78___ is the branch of mathematics dealing with limits and related theories , such as differentiation , integration , measure , infinite series , and analytic functions . A mathematical problem is a problem that is amenable to being represented , analyzed , and possibly solved , with the methods of ___MASK65___ . This can be a real - world problem , such as ___MASK79___ the orbits of the planets in the solar system , or a problem of a more abstract ___MASK13___ , such as Hilbert ' s problems .< br > It can also be a problem referring to the ___MASK13___ of ___MASK65___ itself , such as Russell ' s Paradox . The Palatinate Forest , sometimes also called the Palatine Forest , is a low - mountain region in southwestern Germany , located in the Palatinate in the state of Rhineland - Palatinate . The forest is a designated ___MASK13___ park covering 1 , 771 km and its highest elevation is the Kalmit ( 673 m ). ___MASK53___ , traditionally known as the theory of functions of a complex variable , is the branch of ___MASK78___ that investigates functions of complex numbers . It is useful in many branches of mathematics , including algebraic ___MASK28___ , number theory , analytic combinatorics , applied mathematics ; as well as in physics , including hydrodynamics and thermodynamics and also in engineering fields such as nuclear , aerospace , mechanical and electrical engineering . A ___MASK94___ is someone who uses an extensive knowledge of ___MASK65___ in his / her work , typically to solve mathematical problems .