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Eugène Charles Catalan
Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic minimal surface in the space \mathbb^3; stating the famous Catalan's conjecture, which was eventually proved in 2002; and, introducing the Catalan number to solve a combinatorial problem. Biography Catalan was born in Bruges (now in Belgium, then under Dutch rule even though the Kingdom of the Netherlands had not yet been formally instituted), the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville (1833). In December 1834 he was expelled along with most of the students in his year for political reasons; he resumed his studies in January 1835, graduated that summer, and went on to teach at Châlons-sur-Marne. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruges
Bruges ( , nl, Brugge ) is the capital and largest City status in Belgium, city of the Provinces of Belgium, province of West Flanders in the Flemish Region of Belgium, in the northwest of the country, and the sixth-largest city of the country by population. The area of the whole city amounts to more than 13,840 hectares (138.4 km2; 53.44 sq miles), including 1,075 hectares off the coast, at Zeebrugge (from , meaning 'Bruges by the Sea'). The historic city centre is a prominent World Heritage Site of UNESCO. It is oval in shape and about 430 hectares in size. The city's total population is 117,073 (1 January 2008),Statistics Belgium; ''Population de droit par commune au 1 janvier 2008'' (excel-file) Population of all municipalities in Belgium, as of 1 ... [...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] |
Catalan's Triangle
In combinatorial mathematics, Catalan's triangle is a number triangle whose entries C(n,k) give the number of strings consisting of ''n'' X's and ''k'' Y's such that no initial segment of the string has more Y's than X's. It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan. Bailey shows that C(n,k) satisfy the following properties: # C(n,0)=1 \text n\geq 0 . # C(n,1)=n \text n\geq 1 . # C(n+1,k)=C(n+1,k-1)+C(n,k) \text 1 n it is impossible to form a path that does not cross the constraint, i.e. C_(n,k)= 0 . (3) when m\leq k\leq n+m-1 , then C_(n,k) is the number of 'red' paths \left(\begin n+k\\ k \end\right) minus the number of 'yellow' paths that cross the constraint, i.e. \left(\begin (n+m)+(k-m)\\ k-m \end\right) = \left(\begin n+k\\ k-m \end\right). Therefore the number of paths from (0,0) to (k, n) that do not cross the constraint n - k + m - 1 = 0 is as indicated in the formula in the previous section "''Generalization''". Proof 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Pseudoprime
In mathematics, a Catalan pseudoprime is an odd composite number ''n'' satisfying the congruence : (-1)^ \cdot C_ \equiv 2 \pmod n, where ''Cm'' denotes the ''m''-th Catalan number. The congruence also holds for every odd prime number ''n'' that justifies the name pseudoprimes for composite numbers ''n'' satisfying it. Properties The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 with the latter two being squares of Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...s. In general, if ''p'' is a Wieferich prime, then ''p''2 is a Catalan pseudoprime. References * Catalan pseudoprimes Research in Scientific Computing in Undergraduate Education. {{Classes of natural numbers Pseudoprimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan's Problem
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preda Mihăilescu
Preda V. Mihăilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture. Biography Born in Bucharest,Stewart 2013 he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving a PhD from ETH Zürich in 1997. His PhD thesis, titled ''Cyclotomy of rings and primality testing'', was written under the direction of Erwin Engeler and Hendrik Lenstra. For several years, he did research at the University of Paderborn, Germany. Since 2005, he has held a professorship at the University of Göttingen. Major research In 2002, Mihăilescu proved Catalan's conjecture.Bilu et al. 2014. This number-theoretical conjecture, formulated by the French and Belgian mathematician Eugène Charles Catalan in 1844, had stood unresolved for 158 years. Mihăilescu's proof appeared in ''Crelle's Journal ''Crelle's Journal'', or just ''Crel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Romania
Romania ( ; ro, România ) is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern, and Southeast Europe, Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, Serbia to the southwest, Moldova to the east, and the Black Sea to the southeast. It has a predominantly Temperate climate, temperate-continental climate, and an area of , with a population of around 19 million. Romania is the List of European countries by area, twelfth-largest country in Europe and the List of European Union member states by population, sixth-most populous member state of the European Union. Its capital and largest city is Bucharest, followed by Iași, Cluj-Napoca, Timișoara, Constanța, Craiova, Brașov, and Galați. The Danube, Europe's second-longest river, rises in Germany's Black Forest and flows in a southeasterly direction for , before emptying into Romania's Danube Delta. The Carpathian Mountains, which cross Roma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Royal Academies For Science And The Arts Of Belgium
The Royal Academies for Science and the Arts of Belgium (RASAB) is a non-governmental association which promotes and organises science and the arts in Belgium by coordinating the national and international activities of its constituent academies such as the National Scientific Committees and the representation of Belgium in international scientific organisations. RASAB was formed as a non-profit organization (Association without lucrative purpose) in 2001 by the Dutch-speaking academy KVAB ( Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten i.e. ''Royal Flemish Academy of Belgium for Science and the Arts'') and by the French-speaking academy ARB ( i.e. ''The Royal Academy of Science, Letters and Fine Arts of Belgium''). The association is headquartered in the buildings of the former Royal Stables at the Academy Palace, Hertogsstraat 1 Rue Ducale B-1000 Brussels. History Academies RASAB was founded in 2001 by the two Belgian academies which are connecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Liège
The University of Liège (french: Université de Liège), or ULiège, is a major public university of the French Community of Belgium based in Liège, Wallonia, Belgium. Its official language is French. As of 2020, ULiège is ranked in the 301–350 category worldwide according to ''Times Higher Education'', 451st by ''QS World University Rankings'', and between the 201st and 300th place by the '' Academic Ranking of World Universities''. More than 2,000 people, including academics, scientists and technicians, are involved in research of a wide variety of subjects from basic research to applied research. History The university was founded in 1817 by William I of the Netherlands, then King of the United Kingdom of the Netherlands, and by his Minister of Education, Anton Reinhard Falck. The foundation of the university was the result of a long intellectual tradition which dates back to the origins of the Prince-Bishopric of Liège. Beginning in the eleventh century, the influenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1848 Revolution
The Revolutions of 1848, known in some countries as the Springtime of the Peoples or the Springtime of Nations, were a series of political upheavals throughout Europe starting in 1848. It remains the most widespread revolutionary wave in European history to date. The revolutions were essentially Democracy, democratic and Liberalism, liberal in nature, with the aim of removing the old Monarchy, monarchical structures and creating independent nation-states, as envisioned by romantic nationalism. The revolutions spread across Europe after an initial revolution began in French Revolution of 1848, France in February. Over 50 countries were affected, but with no significant coordination or cooperation among their respective revolutionaries. Some of the major contributing factors were widespread dissatisfaction with political leadership, demands for more participation (decision making), participation in government and democracy, demands for freedom of the press, other demands made by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Châlons-en-Champagne
Châlons-en-Champagne () is a city in the Grand Est region of France. It is the capital of the department of Marne, despite being only a quarter the size of the city of Reims. Formerly called Châlons-sur-Marne, the city was officially renamed in 1998. It should not be confused with the Burgundian town of Chalon-sur-Saône. History Châlons is conjectured to be the site of several battles including the Battle of Châlons fought in 274 between Roman Emperor Aurelian and Emperor Tetricus I of the Gallic Empire. The Catalaunian Fields was the site of the battle of Châlons in 451 which turned back the westward advance of Attila. It is the setting of the last operetta of Johann Strauss II, ''Die Göttin der Vernunft (The Goddess of Reason)'', (1897) and is mentioned in, “It’s the Great Pumpkin, Charlie Brown,” as Snoopy’s crash site after doing battle with the Red Baron. Plan de la cathedrale Châlons-sur-Marne 1859 Archives nationales France.jpg, Châlons en Cham ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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